He/She couldn't do that. An Executive Order cannot undo existing law see: Youngstown Sheet & Tube Co. v. Sawyer, 343 US 579 (1952) . BUT, he/she could order the DoJ to drop its legal defense of Obamacare. And in fact, if the new president felt that Obamacare was unconstitutional, it would be their moral imperative to not defend it, in a court of law.
Au contraire. He/she most certainly could do that. Nobody can force a President to execute a law he/she considers unconstitutional. A president can proclaim a law unconstitutional and refuse to give it any effect. The only recourse is impeachment, which isn't going to happen if most people agree with the President's substantive position. Youngstown Sheet and Tube isn't on point at all. Even it it were, it's just a court case. Court's are powerless to give their decrees any effect. If the President can convince most voters that his view of the Constitution is correct, he/she can act in accord with that view and there is nothing anyone can do about it.
As a Clintonista once observed, “stroke of the pen, law of the land — Kinda cool.”