Will I standing on earth get electrocuted when only touching one end of a floating high-voltage generator?

The answer depends on exactly what the open end is like, and also on the capacitance of the generator.

The Earth has a large capacitance. This means it does not necessarily need to be connected in a circuit in order to have charge flow onto or off it. It can simply charge up a bit.

If the generator also has a large capacitance, then, similarly, charge can flow from it even in the absence of a complete circuit.

If the generator has a small capacitance and the open end is truly open, very well insulated from the earth, then the large voltage will appear between the generator and the open end. There will not be a large voltage between the generator and the Earth, so the man will not get a shock. Something similar to this happens when birds land on high voltage cables. They are mostly ok, unless, in very damp conditions, there is a path to ground through the air. In that example the whole bird is raised to the voltage of the cable, which is not like your thought-experiment, but otherwise it is similar.

If the open end only appears to be open but really has a path to ground, say through the air or along the surface of some material which is insulating but can support surface currents, then things get dangerous. You do not need much current in order to create a dangerous shock, so for a high enough voltage at the generator even a high resistance path to ground will be enough to cause a dangerous situation. For example, many things said to be insulated from ground really only have a 1 to 10 megaohm resistance to ground. So a 10 megavolt source would cause a 1 to 10 amp current. That's a huge current. I have not looked up the details but I suspect even a milliamp would be dangerous. The lesson of this is: in a real experiment do not trust the concept of an "open" circuit at high voltage. It may be that no circuit is really open: all have some finite, not infinite, resistance.