Is it right to say "Rather than GOING OUT, he chose to stay at home" or should it be "go out"?

Billy Bean has missed the point - it is a question of matching the tense/voice.

Rather than go out he chose to stay at home.

Rather than going out, he is staying at home. or Rather than going out, he will be staying at home

You can use either, though "going out" is better as it has the connotation as in "going out for the evening" rather than "go out" which might just mean to leave the room or house.

Rather than go out, he chose to stay home.

Or...

Rather than going out, he is staying home.

You can have either.

If you want to be picky (and who doesn't?) the first form might suggest going in and out repeatedly as a continuous action, and the second a single event, but I guess it would depend on the context.

go out

Either one is correct.