In the United States, governing power is divided among the executive, legislative, and judicial branches. Would Machiavelli have approved of this separation of powers?

No, he would not

Explanation:

Machiavelli authored a popular books called the Prince. In that books, he presented his view regarding how the government should be operated.

He believed that the leader of the country should have absolute authority over the rest of the country. If the king told the people to do something , they have to obey it without any possibilities of fighting back.

The division of government into executive, legislative, and judicial branches contradict this principals. The division was created to Make the leaders of the country have lesser power. (for example, if the president want to do something unconstitutional, judicial branch can prevent it from happening)

in an absolute  monarchy, the  monarchrules as an autocrat, with absolutepower  over the state and government — for example, the right to rule by decree, promulgate laws, and impose punishments. the  monarch isnominally sovereign but the electorate, through their legislature, exercise political sovereignty.