Ok, let's imagine for a second that there were no stars and no planets. How would you then pretend to prove that the Earth is a spinning ball planet orbiting around the Sun, without assuming that the Sun is a space ball bigger than Earth, millions of miles away?
Ok...that's not even close to being an adequate response to the question I asked, but it's an interesting thought experiment. So, no stars, no planets, I'll throw in no moon as well. Just a blank day / night sky, with the sun as it appears to us now.
So, what could we deduce from surface-based experiments?
1. Survey the earth, so we can know accurate positions of places. This would be the first clue that we are on something spherical, as the distances between known places on large land masses don't make sense on a flat earth.
2. Take observations of the sun throughout the day from different places on the earth. This will establish the 24-hour cycle, as well as the annual cycle across the year, and the relationship between the sun's position in the sky and its variation with position - ie how it appears to somebody in, say South Africa compared to somebody looking at it from Europe. In London right now, for example, the sun is about 46 degrees above the horizon, at around 120 degrees true azimuth. In Cape Town right now it's about 20 degrees above the horizon on a heading of about 50 degrees true.
3. Observe gyroscopes, ring lasers, and large pendulums, and note how their behaviour varies with position. Orientated level to a local observer, they show a rotation rate of 15 degrees per hour x sine of the local latitude.
4. The surveyed position data, combined with the rotation data and the observed sun position can only make sense on a rotating sphere. This also is also supported by the behaviour of weather systems, whose rotation in relation to pressure gradients at different latitudes in the two hemispheres only makes sense on a rotating earth. Tides also support this, although without a moon our tidal patterns would be very different.
5. The next challenge is to look at seasonal variations. The best explanation for this is a tilted earth. That would then require that the earth is either rocking back forth towards and away from the sun, or the earth and/or sun are in motion.
6. We can dismiss the rocking because a) that would require an energy source and b) we would detect the motion through gyroscopes, and yet we don't.
7. The tricky part, to get to your point, is whether we would be able to tell whether the sun was orbiting the earth, or whether the earth was orbiting the sun. I'm not sure that you could tell...but more importantly, I'm not sure there would actually be an answer. There is no fixed datum in space - things only move relative to one another, so if there were only two bodies, which one is orbiting the other is somewhat arbitrary.
8. This brings us back to my original point. It is the observations of other things - planets, their moons, our moon, and the stars, and indeed things such as tidal patterns that all sums to together to form a picture of a series of planets in orbit around the sun. How else would you explain any of that in a geocentric system? Could you even begin to draw a geocentric diagram of the earth, other planets and the sun that would correspond to our observations?