the President has veto power and can overturn any federal law after it has passed Congress.
A president can't overturn a law once it has passed. Once a bill has passed into law, it's law. The president does have the ability to veto a bill before it is considered passed. Congress may then override the veto with a two thirds supermajority of each chamber. This is difficult but not impossible.
A president has no ability to veto a constitutional amendment, which many would consider included in "any federal law".
The recent travel ban tells me that the President can use Executive Orders to make anything he wants law without going through Congress.
While presidents have a lot of power with regards to executive orders, they can't actually make anything they want law.
The travel pause is based on a section of 1952 legislation allowing the president to control immigration for national security.
Other executive orders use presidential discretion in regulation, etc.
And of course, some executive orders are unlawful and overturned via judicial review.
If the President has both full negative and full positive power over the law, does that not make him a monarch?
Even if we allowed that presidents had full negative and positive power over the law, I still don't believe that would make them monarchs. Monarchs are typically lifetime positions that are either inherited or selected from an inherited group.
Monarchs don't necessarily have full power over the law. Many are essentially figureheads. For example, technically speaking the Queen of England has to sign legislation for it to become law. But as a practical matter, she never fails to do so. So monarchs may have less power than a president.
The president is certainly the head of state in the United States. But since it is an elected, temporary position open to any natural born citizen of sufficient age, I do not believe that it is the equivalent of a monarchy. And it wouldn't be even if the presidency was otherwise more powerful. Perhaps it would be a dictatorship or an autocracy, but not a monarchy. Of course, from dictatorship to monarchy is not a big leap.
At minimum I would say that a monarch has to be limited to some group via heredity. Almost all have lifetime terms, and most are selected purely by primogeniture. But there are exceptions to those rules.