An equivalent question is “can you turn a globe so all you see is water”.
The answer is no, but take a look at the globe at this site. It’s not that far off. There is certainly an angle where 90+% of the visible earth is ocean.
Good point. If we move away from the December solstice, we stop wasting daylight on the southern Pacific Ocean and put mire of it on northern Asia. Here’s late October, which looks to have a good bit more land lit up.
I believe to atmospheric refraction the sun can appear 50 minutes higher on the horizon than its actual position So land masses can be up to 181 deg 40 min apart and see the sun at the same time.
That’s true when you look at surface area, but in terms of longitude, the OP’s claim that the Pacific covers a little less than half of the Earth isn’t that far off. The easternmost point of mainland China along the East China Sea is at about 122 degrees east; the westernmost point of the West Coast of the contiguous United States is at about 124 degrees west. That gives you a longitudinal span between these two points of 114 degrees, which is less than half the circumference of the Earth. You can turn the globe on Google Earth such that both points are within view at the same time.
It is true atmospheric refraction extends the amount of earth from which the sun is visible. Also, the sun is quite visibly not a point source. This adds a little more than 30 minutes of arc as well.
But still, in 2019, there’s sufficient landmass distributed across the globe that there is never a date and time when the sun is visible from every bit of land, even after excluding tiny Pacific Ocean atolls.
It doesn’t answer this question, since the sun never reaches zenith at that location; but the center of the land hemisphere seems to be somewhere close to Nantes, France.
Though on top of that, you have to ask whether that angle is one that is possible to be facing away from the sun at any point in the year.
Except that Antarctica + Australia combined have lower area than North + South America, and the times you linked (including in October) have both Americas in the dark. So my guess is that summer solstice might be better.
The website you linked is pretty neat - seems like you can get pretty much all of the Americas and continental eurasia lit up at once (~11:15 UTC in June), but not sure if that actually corresponds to maximum land lit - there may be some optimizing to increase lighting in Antarctica at the expense of the fringes of Alaska/Russia.
The above illustrates the effects of refraction pretty well since the E Asia/S America antipodes are both (barely) lit.
While refraction might play a role, you should also note that the sun isn’t actually a point source: it covers an angular width of about 0.5 degrees.
Any usable sunrise/sunset calculator must account for both atmospheric refraction and the diameter of the sun, which will cause it to occur when the centre of the sun’s disk is about 50 minutes of arc below the horizon. Parallax is small and can be ignored, unlike the case of the moon.