But he probably knows more about this than you do.

It is not the conclusion of science that when two objects are in motion around each other that it is always equally valid to say that A orbits B as it is to say that B orbits A. There are physical effects of rotation that are easily measured, and others have mentioned some. (Foucault's pendulum, the Jet Stream, coriolis forces...)

For a simple, common sense illustration of this, pick up a bucket of water and whirl it around your head. Ask whether you are going around the bucket or the bucket is going around you. His claim is that the two statements are equivalent, and clearly they are not.

Now, that said, there is some truth that he is mangling. That truth is that the laws of physics hold regardless of the coordinate system you use to describe what is going on. So you can pick any coordinate system that you choose, then be really careful about how you state the physics, and you can make all of the answers come out right. (To do it exactly right you need to introduce the concept of a "metric". There is no need to worry about the details though, suffice it to say that it can be done.)

But the coordinate system is completely and utterly meaningless! Switching coordinate systems is like switching between km and miles, it has absolutely nothing to do with what is actually happening, it just changes the language that we use to talk about it. The coordinate system is free to be changed exactly because it is meaningless.

For instance it is possible to measure how much an object is rotating or accelerating. This measurement does not depend on the coordinate system. Guess what! You find that the Earth rotates about once a day! (That's off by a few percent because in a day we also move partway through our orbit.) By contrast you find that there is a coordinate system that is still relative to the center of the Sun which is barely rotating at all! And in that coordinate system, the Earth goes around the Sun once a year. (And rotates about its axis a bunch more times.)

Furthermore he is wrong to say that this requires general relativity. The above fact is is a triviality that has been noticed since Rene Descartes invented Cartesian coordinates. When Foucault came up with his pendulum, he used the pendulum as a way of measuring a component of the rotation of a rotating coordinate system.

So why does he think that general relativity has anything to do with this? Well in general relativity once you introduce masses, there is no perfect coordinate system. In theories up to and including special relativity there are "inertial frames of reference", which is to say simple Cartesian coordinate systems in which objects that are still relative to the coordinate system stay still unless something pushes them. Generally people prefer working in these coordinate systems because the laws of physics get a lot simpler. But in general relativity there are no inertial frames of reference. You have to deal with the non-inertiality of your frame of reference.

However while no reference frame is perfect in general relativity, some reference frames are closer to being inertial than others. Working in those is still preferred, because the corrections to treating physics naively are much smaller and more manageable.

And, as you might guess, a geocentric coordinate system is a lot farther from being inertial than a heliocentric one.

Cheers,
Ben