The entire thing doesn't seem to be the oath, just specific parts of it. And most of it isn't unique to the Enterprise, it's part of the Federation charter and Starfleet's general orders.
Seeking out new life
JANEWAY: By adhering to the oath you took as Starfleet officers to seek out life, not destroy it.
Boldly going places
Mudd: The moment you decided to boldly go where no one had gone before.
Seeking out new civilisations
Janeway: And as the only Starfleet vessel assigned to the Delta Quadrant, we'll continue to follow our directive to seek out new worlds and explore space.
