Mutt joining the Navy might take him too closely down the James Bond route (Commander Bond, Naval Intelligence etc), and into a whole career that takes him away from the archaeologicial routes of Indiana Jones.
I've always considered Indiana to be more a civilian volunteer - he volunteered with the Belgian army and became a captain during extraordinary circumstances (the battlefields of the First World War), but he got out back into arachaeology. Before the US entered the Second World War Indiana was undertaking aracheogical digs, sometimes in enemy held territory. His skills and his previous asssociation with US military intelligence (as in Raiders), lead to him working for the military after 1941, alongside Mac. It strikes me that this was merely a product of extraordinary circumstances again, not a desire to enter a career with the military.
I think the over-riding influence from father to son should be a love of history and archaeology, rather than military service. In Indy's eyes Mutt would earn the hat if he shared his love of uncovering the past.
To become a Seal would involve incredible commitment for Mutt, and would hamper his studies. He already has a lot of catching up to do on his journey from rebel to student to teacher. Much of the enjoyment derived from watching Indiana Jones movies is that for much of the time Indy's making things up as he goes along. In combat he's a dirty fighter, not a professional assassin.
If Mutt is to earn the hat I'd like to see him as similar to his father, otherwise future films might go too far the way of James Bond. Much of the charm of Indy is that he is often the underdog, doesn't always win, and resorts to underhand tactics to gain advantage. I like to see Indy 'winging it' rather than systematically planning his operations, as a member of the special forces would be compelled to do.