The Lockheed Martin Atlas V and Northrop Grumman Antares are the only American large launch vehicle that still use Russian-built engines. The Atlas V uses the RD-180 built by NPO Energomash for the first stage, and previously, the Orbital Sciences Taurus II/Antares rocket used the Aerojet AJ-26 which is a refurbished and modified NK-33 in the first stage, but now uses a downrated RD-181. The last launch of Atlas V is scheduled for 2022, after which it will be replaced by the Vulcan using a Blue Origin engine, while the Antares is is only used for NASA Commercial Resupply Services program. Since that program is only to support International Space Station (ISS) logistics and it is now questionable that the ISS will continue in service until the planned retirement in 2030 without Russian partnership, it probably isn’t the issue that it might seem to be. I don’t know how many engines that Northrop already has in inventory but it is probably sufficient to cover the CRS-2 flights through 2023, after which additional bids are still open to other CRS-2 contractors (SpaceX, Sierra Space (formerly Sierra Nevada Space)). I would assume that SpaceX would be pleased to pick up more missions provided they can fit it into their launch tempo, while Sierra has yet to launch and is predicated on a successful qualification of Vulcan in 2023.
The “U.S. rocket engine-building industry” in general is not doing great. The heritage of legacy liquid engine manufacturers (Rocketdyne, Pratt & Whitney, Aerojet General, TRW) has pretty much been lost through attrition. There is still some native knowledge in building cryogenic upper stages and small thrusters by the inheritors of these companies but building large primary booster stage engines is a complicated activity requiring many different engineering disciplines and a lot of empirical knowledge that has not been captured by another generation. The same is true for solid rocket motors (Thiokol/ATK, Aerojet, P&W CSD, Lockheed Propulsion) to the point that only Northrop Grumman (which purchased ATK/Orbital) still has the knowledge and capacity to build large solid motors, and their production is mostly limited to the US Navy Fleet Ballistic Missile (FBM) program (i.e. the D-5 ‘Trident II’ first and second stage).
SpaceX, of course, builds engines for their own use and is actively engaged on an ambitious plan to build very large, high thrust methane/LOX full flow combined cycle engines, but Elon Musk and Gwyenne Shotwell have made it clear that they have no intentions of selling engines to other launch vehicle manufacturers or integrators, so while that is good for SpaceX it doesn’t benefit competitiveness in the spaceflight industry as a whole. Blue Origin has been developing and building engines (most notably the BE-4 used on Vulcan) but their development has been slower than expected and halting, indicating problems during development which is entirely expected. Beyond that there are a number of companies building smaller liquid and hybrid rocket engines for smallsat/nanosat launch vehicles using a variety of approaches but it remains to be seen how many will be successful, and scaling up a small engine to much higher thrust is a non-trivial exercise; the features that work in small engines often cause fundamental stability problems in larger combustion chambers as well as the scale of heat flow and problems with propellant delivery, et cetera.
I would say that aside from reliance on Russian engines (which was problematic but necessary at the time; attempts to produce a licensed version of the RD-180 by Pratt & Whitney resulting in a setting a billion dollars on fire without successful production) the United States has a lot of problems in just sourcing materials and critical components because even if we have the ability to produce specialty components there are thousands of other materials and components that are ultimately commercially sources from foreign suppliers, including nearly all mechanical fasteners, ‘normal’ commercial steels, carbon fiber and resin precursors, et cetera. A purely domestic supply chain is really important for being able to maintain a secure rocket manufacturing capability and we no longer have anything like the capacity we had even circa 1990 when globalization was already in progress.
I know a lot of people like to think that we can just ‘3D print’ everything that is needed but in fact components like valves, fasteners, sensors, microprocessors, et cetera, are all things that have an enormous and mostly invisible infrastructure of supply chain logistics, knowledge base, materials, et cetera behind them that would need to be reestablished to have a truly domestic supply of rocket engines and other critical flight structures and components that could not be compromised by disruptions in the global supply chain system. It has become a major issue with the aforementioned FBM system (because domestic sourcing of materials is required and has to go through a lot of gyrations to make things like Toray fiber a ‘domestic’ product) and with the upcoming development of the Ground-Based Strategic Deterrent (the long-overdue replacement for the Minuteman III ICBM) it may actually become a show-stopper without a lot of waivers, and even then a lot of foreign suppliers have severe restrictions when it comes to supplying materials used in the manufacture of weapons of mass destruction. So it is generally a huge problem even beyond the loss of heritage knowledge and manufacturing experience with engines and motors.