I don't think that follows at all. If anything, the free flow of information that the Internet has allowed has resulted in the opposite of cultural uniformity, because once-isolated fringe ideas are now able to get broader hearings and larger followings. And historically, the parts of the world where different ideas have been free to mix and cross-pollinate, far from being melting pots that merged into homogeneity, have instead been the birthplaces of multiple new, competing philosophies and religions, as the different ideas have mixed and matched and clashed and reacted in a variety of different ways. When people have that many possible paths to choose from, there's no way in hell they're all going to mutually agree to head in the same direction. Cultural uniformity is only found in isolated areas where exposure to new ideas is limited.
The free flow of information hasn't led to cultural uniformity but it has led to secularism, tolerance and cultural cross-pollenation. The US and Japan have been stealing each others' ideas for 200 years. Kurosawa was influenced by the West, then Sergio Leone and George Lucas came along and blatantly copied Kurosawa. It may seem like a lot of religious governments are just harshly cracking down on dissent, but it's only because they saw the civil rights and equality debate starting to happen and got scared.
So you're right, we would not have one uniform culture throughout the whole galaxy. We might see lots of different cultures who mutually tolerate each other and copy each others' best ideas, who share a belief in individual freedoms and have open, friendly lines of communication to pursue common goals.