In "Standard" American, Is It Possible to Bid the Opposing Suit "Naturally" Over a Takeout Double?
In "standard" american (or typical variants of 2/1) it is normal to play 2H in the sequence: (1H) - X - (P) - 2H, as a close to game forcing hand (opposite a minimum takeout double) or better. There might be variants which different partnerships play, but it is almost never a natural bid.

2H should not be to play, as it makes no sense to play in 2H there. You are contracting to make at least 8 tricks with hearts as trump, with partner showing shortness in hearts by his takeout double! If you make those same 8+ tricks, opposing a 1H doubled contract, don't you rate to make at least +200 (or more if opps are vul)? There might be cases where you let 1H doubled make, but in that case, you are probably down 2 in your own 2H anyway and those would be very rare.

Over a minor suit takeout, some people might consider playing it as natural, as opponents could have shortness there and might have the option to try and run to a longer suit, but I believe the usual is still to play 2D in: (1D) - X - (P) - 2D, as close to game forcing hand etc and not natural.