Kinda like this video: https://www.youtube.com/watch?v=QXMhRp_laHc
But with the laser ground-mounted instead of projectile-mounted, for weight and power reasons.
Yes, that is why I said, “such a method.”
Vaporize the atmosphere to what? It’s already vapor!
It’s not as if material magically disappears when you shoot a laser at it. As my above Youtibes from Lewik Myrabo’s work show, focusing on atmosphere does make it expand, not form a vacuum. It’;ll actually push backwards towards the projectile (although it’s far enough away that it would have negligible effect).
While we’re talking about getting into space, this is a problem if we’re looking to go into orbit as a satellite. That said, it’s a different problem if we want to, say, head for Mars. Aim your launching method near the moon and launch far enough and you can use the pull of the moon to alter your initial trajectory, and/or land on the moon then use the slingshot someone else suggested to launch back off the moon. The orbital trajectory has to be acted upon by an outside force to solve the problem of the launch point intersection, but that outside force doesn’t have to be generated by the projectile itself…
Which was a fictionalized version of a late-40s concept that came out of Los Alamos, Project Orion.
I admit that this part of my response was based more on intuition than physics, as opposed to the second part, and in thinking it over, I see that you’re right. Even if you shot the projectile horizontally, it would go around one orbit and come back to hit you in the ass! 
I also wondered if the rotation of the earth could be used to advantage, but that doesn’t work, either. I remember playing with some orbital simulators and getting it impressed on me that whatever you do to an orbit, at the point that the force stops the ballistic path always comes back to where it started. Now I feel dumb. :o
Somewhat related – if only peripherally – to the last post: I’ve always found it remarkable that if the collision theory of the moon’s formation is correct, that the collision debris (which was, in effect, “shot” from the surface of the earth) not only managed to go into orbit, but established an orbit that was almost perfectly circular – apparently completely spontaneously.
Perhaps just ablating the base* of the projectile with the ground-based laser would be more practical, to circularize it’s orbit.
“How did I put the thing into space? A cannon! How did I put it into orbit? A laser cannon!”
*Well, “part of the projectile currently pointed retrograde,” anyway. Perhaps itmight be what was left of a protective “nose cone” from the launch, rotated back to face the laser battery.
Kerbal Space Program gives one a good idea of what is necessary to circularize an orbit in space. 
Saturn appears to be forming a new moon from ring-stuff: most likely, the collision cast up a massive debris field that coalesced into our moon, hydrostatic equilibrium pulling it into a sphere. Consider, also, when you look at the moon’s face, you can see evidence of thousands of big and small adjustments having been made to its orbit.
And maybe “spontaneously” is ever so slightly misleading, as we generally do not frame spontaneity in terms of scores of millions of years.
Speaking of the moon, it seems to me to be the wild-card: a really good mathematician could chalk up a projectile path that might be bent into a circular orbit via lunar influence.
As Francis Vaughn noted, any object in a closed ballistic (unpowered) orbit will return to its origin (subject to precession of the orbit and perturbations of gravity, of course). Now, it would be possible to launch an object in such a way that the drag would shape the trajectory such that it does enter a very low perigee, highly elliptical orbit, and with the right impulse and precision it might even remain in orbit for a few passes, but it will be so low that the drag will shortly bring it back down. If you want to enter a stable orbit or one that is mostly circular, you have to thrust sideways to gravity at some point during your ascent in order to achieve enough tangential momentum to remain in orbit. Of course, with enough energy you can launch something into space into a suborbital trajectory, and with even more into an escape trajectory, though in both cases you will find that the aeroloading and aeroheating of accelerating an object to such a speed in the lower atmosphere will destroy any real material. (The “manhole cover” from the Plumbob Pascal-B shot very likely vaporized due to aeroheating before reaching the Karman line.)
As noted, the atmosphere is already “vaporized” in its native state, and adding more heat would result in greater pressure, not less, nor would the mass of air decrease; essentially you’d be adding the energy from the laser to that of aeroheating from compression. A laser sufficiently energetic enough to disrupt air (e.g. dissociate it into atomic oxygen and nitrogen) would do so from the ground up.
Luna makes a poor gravitational ‘slingshot’ to go to Mars because there are very few opportunities to launch directly from Earth and around the Moon that result in a trajectory that is directed in the plane of the solar ecliptic, owing to the fact that the Moon is inclined over five degrees from the ecliptic. (Technically, we’re more concerned about the invariable plane than the ecliptic, but the results are about the same; going around the Moon directly from Earth and thence outward will typically result in kicking the spacecraft out of the plane of the solar system, which requires more energy to correct than is gained by the paltry amount of momentum picked up from the Moon.) In short, this is not a viable option which is why it is not used for interplanetary probes to Mars or other planets.
The debris ejected from the Earth that eventually formed the Moon was not one continuous mass but was a enormous cloud of molten material, most of which fell back to Earth. That which remained formed an accretion disk around the Earth which eventually coalesced into Luna and other smaller bodies which still orbit in pseudo-chaotic trajectories as Near Earth Objects. That the orbit is almost circular is the least remarkable thing about it; any natural collection of diffuse material that coalesces will tend to fall into a circular orbit just because it has the least perturbation, and all other mass either eventually combines with it or is ejected from the system. There is nothing “spontaneous” about it; the result is just simple Lagrangian mechanics.
Stranger
Another way of stating this: Most of the effects that will alter an orbit will do so by lowering the orbital speed, and most of these effects are strongest at the periapsis (closest point). Lowering your speed at the point of periapsis will lower your apoapsis (highest point), but leave your periapsis point unchanged. Thus, orbits tend to circularize.
Thanks to Chronos and Stranger for the explanation. I would say, though, to Stranger and eschereal, that my use of the word “spontaneous” was in the accepted sense of “arising from internal cause: resulting from internal or natural processes, with no apparent external influence” which is surely a correct description of how the post-impact debris coalesced into an orbiting moon.
That’s interesting. Could a volcanic eruption propel objects that far?
As a matter of fact, there is the “Verneshot” hypothesis.
Only on a planet with no atmosphere and only in a single body system. In theory, you could use a moon slingshot to circularize your orbit instead of rockets.
Gerald Bull’s HARP was launching small shells into the upper atmosphere. That’s a far cry from orbital speeds. To orbit the earth not far above the atmosphere requires approximately 18,000 mph; consider the amount that drag may slow a vehicle just after the launch, and you need to give the launch quite a large velocity.
One theory was to build a linear accelerator climbing the side of a very tall mountain (preferably near the equator where you add the earth’s rotational velocity, 25,000miles/24hrs = 2,000mph
The basic formula is v^2=2ad
Pick a final velocity, say 16000mph =23,466feet/sec. Pick a survivable acceleration, say 10g = 320ft/sec^2.
I get the answer is that you need to accelerate for 860,444 feet to get up to speed - 162 miles. Remarkably, at 10G that would be a 73 second ride. (v=at)
Use 15G acceleration and you’re down to 108 mile long magnetic accelerator. So- something that can accelerate to orbital speed in a realistic distance - say 20 miles - would basically pulp the occupants. It would only be good for solid small projectiles.
The linear magnetic launcher concept works best in vacuum (no air resistance) and also, the concepts kicked around in SciFi usually meant instead of imparting orbital velocity, the launcher would replace the work of, say, the first two stages of a Saturn-type vehicle and loft the projectile above the majority of the atmosphere so it did not need a lot of fuel simply to get airborne. (Same idea as launching from a high-flying aircraft with a good head start.
Note that, if you don’t have that pesky atmosphere to deal with, you can launch at any angle: Your accelerator can just be sitting on level ground and shoot horizontally. We only launch rockets vertically because the atmosphere is so problematic, that we want to get out of it as quickly as possible.