The confirmation is along party lines, with 49 Republican senators voting in favour of Burch, 44 Democrats against and 7 senators not taking part in the vote.
Burch previously headed the Trump-affiliated organisation "Catholic Vote", which he founded, and was a clear critic of Pope Francis.
Among other things, he had spoken out against the declaration of blessing "Fiducia supplicans" and accused Francis of vindictiveness after he deposed the Texan Bishop Joseph Strickland.
In an initial statement following his confirmation, Burch expressed his happiness at now being able to take up his post and emphasised the USA's close ties with the Vatican: "The relationship between the Holy See and the United States remains unique in the world, with the global reach and moral witness of the Catholic Church forming an essential part of the USA's efforts for peace and prosperity."
Blocking nominations as leverage in the budget dispute
The Senate was unable to vote as originally scheduled in May after Hawaii Democratic Senator Brian Schatz announced a move to block presidential nominations as a means of pressure to protest against cuts in development aid and the winding down of the United States Agency for International Development (USAID).
As a result of the delayed confirmation, the USA was not represented by an ambassador at the inauguration of Pope Leo XIV.
In the meantime, the US Embassy to the Holy See was headed by a career diplomat with the rank of chargé d'affaires, not an ambassador appointed by the president.
Burch is one of ten personnel appointments that the Senate decided on Saturday as part of a compromise.
More than 100 others remain unconfirmed as Republicans and Democrats have been unable to agree on contentious budget issues.