In fairness to Notre Dame they don't have to face Navy. They choose to. And the reason why they play Navy dates back to WWII. In a rush to educate and graduate more officers for the war the Navy enlisted the help of colleges and universities to train future officers. Notre Dame was one of them. On the flip side, with all the young people enlisting in the armed forces to fight the war colleges and universities were struggling to stay in business because of a lack of tuition, lack of students. Notre Dame in particular was in danger of folding and closing forever. By agreeing to educate and train officers for the Navy during WWII Notre Dame was able to stay open and survive. Ever since Notre Dame has felt they owed Navy a debt for saving the school.
There was talk in the media during the 80's, 90's and 00's up to 2007, when Navy finally beat Notre Dame for the first time in 43 years, to why Notre Dame kept playing Navy. The main reason was because Navy saved Notre Dame. Thats why, when you watch on Saturday, the Notre Dame football team and fans show a lot of respect towards Navy.