ND respects Navy and always wants to help them. This is not a true rivalry on the field, don't kid yourself. If rivalry means you play annually then that is a broad definition and most teams have MANY rivals.
See the list below of rivals based on your criteria (all more interesting to the players and ND fan base):
USC
Michigan
Michigan State
Purdue
Boston College
Stanford
Even...Miami from back in the day.
Old article, but straight from it paraphrases what ND fans believe:
The “rivalry” with Navy isn’t a rivalry at all. Rather, the Irish have a great amount of respect and gratitude towards the Midshipmen. The Irish and the “Middies” have played each other annually since 1927, making it the longest uninterrupted intersectional rivalry in the nation.
Even though the Irish have owned the series between the two schools, there is no animosity between them, and Notre Dame has deep respect for Navy.