No, he cannot sign an executive order to repeal the Affordable Care Act since it's a law. Laws can only be changed or written by Congress.
Executive orders cannot reverse a law passed by Congress. Limited in scope, their sweep isn't as broad as a piece of legislation.
Article I, Section 7 of the US Constitution states that all legislative powers are vested in Congress.
However, the President can weaken the enforcement of the Individual Mandate through executive action. He has in fact already signed an executive order that would weaken enforcement for all Americans to purchase insurance, should it be carried out. One can submit their taxes without filling out the insurance question, thus making enforcement more difficult.
It's worth stressing that this executive order does not grant any new powers or change the wording of the law, it simply weakens enforcement and scale back regulations should the Department of Health and Human Services want to.
Since the EO is vague, it's up to interpretation and enforcement so it would still depend on whether the HHS wants to continue to enforce the Individual Mandate. At the close of this year's tax season, It seems like the Trump administration is still enforcing it.
But Trump’s continued enforcement of the individual mandate — a provision that Republicans have targeted for elimination in their repeal bill — shows the depth of uncertainty within the administration over how to use its regulatory power to weaken the health care law while the GOP struggles to enact a replacement.
(emphasis mine)
The full text of the executive order can be found here.
