No, the NX-01 was named after the Enterprise E that Cochrane saw through his telescope, Deanna told him the name of the ship. Kirk's ship was named after a long line of American naval ships.
If the Enterprise (prime) were named specifically after the NX-01, then wouldn't the NX-01 have been present in the display of former Enterprises in TMP?
Debatably, prior to the events of FC, the NX-01 historically wasn't named Enterprise.
Where is it stated that the NX-01 was named for the NCC 1701-E? The NX-01 has a "legacy" display just like other ships of the name Enterprise. So the implication is it was named or previous ships o that name.
No two legacy displays are the same. There are always "omissions" in every display.