Sorry, but if you aren't married the chances are next to zero. And if you aren't close to his C school and there is traveling involved, that makes it worse.
Consequences? When I was born at the Navy hospital in Pensacola, my dad was on an aircraft carrier in the Mediterranean. That's life in the Navy.
He only gets a week to ten days of regular leave after A school, and overseas travel has to be approved.
Also, paternity leave is only ten days, to be used within a year of the birth of the child at the discretion of his command. Married sailors only, and even of he managed to get to you and marry before the baby was born, he'd not be released from training.
Once he is in C school, he cannot take leave until he graduates, and then only if he has earned enough days, and a lot will depend on his report date to his new command.
If he needs to go overseas…then it's going to be a process. There is some training that he needs to go through, and get a brief. The requirements for the request is going to be based on what country it is. Some countries require a 30 day notification, some a 60 day.
If he is going for a security clearance, he needs to understand that you will be investigated as well…given that you're a foreign national.