This is actually an incorrect view of electron motion around a nucleus. This model (Bohr model) explained some things very well; however, it did not fit completely, in that the model would predict a type of synchrotron radiation due to the reasons you point out. This is not observed.

This led de Broglie to a wave model of the electron that fits much better. Ultimately, this became what we call quantum mechanics.

So to answer your question, the electrons are not "producing" some kind of energy. Though based on the model as you describe it, they should. Even if they did though, I'm not sure what that would have to do with gravity.