Cardona, who was promoted from lieutenant junior grade, follows in his father's footsteps as a member of the Navy.
Cardona, a four-year NFL veteran, has juggled his football responsibilities in Foxborough and active-duty commitments at the Naval Preparatory Academy in nearby Newport, Rhode Island, per ESPN. A fifth-round pick in 2015, Cardona hasn't missed a game in his career.
"[We're] happy to recognize Joe on this day," said Belichick, whose father coached at the Naval Academy for 33 years. "It just makes it a special day for all of us."
"It's pretty emotional when you see someone take the oath like that and raise his hand," Tom Brady told reporters. "It's a big commitment Joe's made, and I think Joe has done it very proudly and does a great job educating a lot of us on what it means to him. I think we all appreciate that.
"We see the work he does here for us, but also the work he does for our country. I've talked to Joe a lot over the years about what his roles and responsibility are and I think everyone is proud of his accomplishments."