Thanks for cleaning and re-checking the backmark, and reporting what it says.
Time-dating your US Navy button:
1- The "plain block" lettering
2- being marked "Prov. R.I." (Providence, Rhode Island)
3- the eagle's head looking toward the viewer's right.
Having those three characteristics IN COMBINATION means your US Navy button dates from sometime between approximately 1900 to as late as early-1943. (In 1943 the Navy changed the emblem on its buttons to having the eagle's head looking leftward instead of rightward.)
If you ever manage to make out the name above the "Prov. R.I." please let us know, because we might be able to give you more-specific information about it.