There is separation of powers to be considered; an imperial presidency is too high a price to pay even to stop H-1B. But the president can certainly bring a lot of influence to bear.
Regardless, there is no stipulation in the US Constitution that we MUST allow immigration into this country. It is solely the purview of the State Department (under direction by the Chief Executive). It is a fictitious right the left has concocted to overcome the fact their bullshit tenets don’t convince Homeland Americans that we need the diversity and richness of immigrants to ‘complete us.’