Does the President of the United States have the power to personally arrest someone? In other words, would it carry the same authority as that of someone in law enforcement?
Probably not what you are looking for but technically yes:
That said the Attorney General is the chief law enforcement officer for the federal government (for federal laws) and one in each state (for that state’s laws). When I see that it means to me they are the top dog when it come to law enforcement and would be the highest ranked official who can arrest someone although in practice I think they tell someone else (local police or sheriff or FBI or whoever) to go do it.
The President, given his powers, probably could exert strong pressure on various law enforcement to have someone arrested (certainly more ability than the Average Joe would) but I doubt he must be obeyed. If the officers directed by the President to go arrest someone thought there was no grounds for it I suspect they could refuse.
According to the Constitution Article II Section 1
It is pretty well established by the Supreme Court that the President as Chief Exacutive does have some unenumerated powers (Myers and Curtiss-Wright), but the extent of these powers is a gray area (for example The Youngstown Case).
All-in-all, I would say that the President is given the power of arrest for federal crimes as part of his official duties, but any such case would clearly end up in the Supreme Court.