Not completely, while the Universe may be infinite spatially, the Big Bang theory posits that there is a finite amount of energy and matter in it.The Universe is infinite however in two of the three probable models.
An open Universe has a negative curvature and is infinite spatially and have no boundary
A closed Universe has a positive curvature, this is the only postulated finite Universe model. This one however is the least likely due mainly to the paradox that wherever you begin, you must end up, much like travelling around the earth in a straight line.
Then we have the Flat Universe which has no curvature and is flat like a sheet of paper, but this type like the Open Universe is also infinite spatially so has no boundary. This is currently the most popular of the postulated Universe models among Physicists.
So due to the fact that the Universe is in all probability infinite, the rest of your conjecture is somewhat questionable.
That along with entropy suggests there isn't going to be an infinity of galaxies.