Not until January 20th. He has to be sworn in first. That's comes from an old trivia trick question.
"What is the maximum amount of time that a president can serve under the provisions of the 22nd Amendment?"
Most would say eight years...and they would be wrong. The answer is 11 years and 364 days. Provided that the president dies, resigns, or is removed from office on January 20th, the vice president becomes president officially on January 21st. They can serve the entirety of that president's term, which will not count to their time in office as they are serving in presidential stead. Then they can run twice over winning both times. Given that unlikely scenario, the longest a president can serve is 11 years and 364 days.