The short answer is no, not immediately upon graduation.
The longer answer is that the Coast Guard Academy is the only academy where cross-commissioning into another branch upon graduating is not allowed. If you go to USCGA, you do your time in the Coast Guard. How do I know? The end of my second class year, I started researching the possibility of crossing over to the Air Force. Because the other services are under DoD, the funds for educations/training come from the same place so it's easier (although still very difficult) to get approved for a branch transfer. At the end of the day, I'm glad I wasn't able to transfer. The Coast Guard has been incredible. However, following your initial service requirement of 5 years, you can then apply for a transfer without ever being discharged from the service.
As TennisDad mentioned, there are Navy billets open to Coasties. As a second tour JO, you can be assigned to a Navy ship to be their Navigator (apparently, they can't do it on their own

). I have a friend who just finished his tour a Navy navigator and he loved it. He'll be the new Commanding Officer of a CG cutter in Hawaii come this summer.
If you really only want to serve in the Navy, go to USNA. If you don't get in, try for a 3 year NROTC scholarship or go to another school and apply again. Apply until you can't anymore and then apply for OCS. Are the two services linked because they are the premier maritime services in the world? Yes. But there are still vast differences in missions, capabilities, career paths etc. that might push you towards one service rather than the other.