The family is "all happy and healthy" a source confirmed to PEOPLE, but the couple won't be releasing their son's name just yet.
Many of Gordon-Levitt's fans weren't aware his girlfriend was pregnant as the actor likes to keep his private life away from the public eye.
Gordon-Levitt, 34 and McCauley tied the knot in a quiet ceremony at their home in December 2014, after dating for a year.
Speaking to PEOPLE in 2013 about his decision to keep things private, he said: "I have a girlfriend but I tend not to really like to talk about it in public.
"The girl that I'm with, she really doesn't want to be a part of that."