Can you get a pension from the US navy after 7 years of service?

Only in special circumstances, and that's typically if you're medically discharged after being wounded, or your duty has caused health issues that leave you unable to continue active service. For all others, 20 years minimum.

Of course there are waivers for any situation, but medical discharges are the most common, and you don't often get full benefits. It depends on the situation of the individual, and for most it's like SSDI.