There are a couple of things that can happen.
You can bring a case to court, and if a federal court finds that the provision is unconstitutional, than it no longer has the force of law.
If congress thinks the president is being particularly abusive of his power, they can impeach him, and potentially remove him from office.
You also have to consider that a lot of bills passed by congress explicitly give the president authorities to make certain decisions, and this might not actually be unconstitutional because of that.
The affordable care act itself gives a lot of authority to the president and members of the president's cabinet.
Just search the bill for the word "secretary" or "president" to find things that the president or a secretary of the cabinet has the authority to do:
Now, bills like this one have numerous amendments and recursive language that makes parsing them harder than one might think, but whenever the president makes an executive order or does something that seems like changing legislation, it's normally because there's a law written by congress that says that he can.