Rome's navy was actually very important when it had to undergo the Punic Wars against Carthage. Being that Carthage was on the other side of the Mediterranean, sea dominance was critical at the time. Since Hannibal had to march his army across Gaul and Hispania (modern day France & Spain) and while he did manage to keep Rome on its toes for a while he simply didn't have the forces to take on Rome's defenses. Critically, if he had the sea power necessary to bring over more units consistently and quickly, Rome may have fell. However, skilful Scipio Africanus managed to land a sizable force at the doorstep of Carthage in North Africa and his key victory in numerous skirmishes near the city caused the Carthaginian's to capitulate to an Armistace. However, afterwards, even during civil wars and conflicts with outsiders; their Mediterranean dominance wouldn't be challenged until the fall of the Empire because most enemies of Rome in the Mediterranean had been subdued. Another potential candidate is the Macedonian Wars in which the Roman's subjugated Greece but from what I've read the naval battles were skirmishes and mostly blockading on the Roman's part. Another major engagement of the Roman Empire was during Antony's Civil War but this was an exception to the peaceful "Roman Lake" that was the Mediterranean.
Some key Battles were the Battle of Actium (Antony's Civil War) & the Battle of Lilybaeum where Rome crushed the Carthaginian Navy and asserted naval dominance.
Rome's navy would be challenged during it's collapse when outside groups like the Goths, Arabs & Vandals (who rose a navy and engaged Rome's) but ultimately the western half was in such a decline that the navy didn't do much to impede them.
As /u/laker_man said, it was very important during Republican Rome's struggle with Carthage, in particular the First Punic War, which was essentially fought primarily at sea after Rome quickly took over the inland parts of Sicily with help from Syracuse, which also controlled a good portion of the island. Some of the more notable battles in this war were Economus which was by far the biggest naval battle of the three Punic Wars, Battle of Mylae where Rome demonstrated the Corvus (Crow), allowing them to pin Roman and Carthaginian ships together allowing the Romans to board much easier, rather than ramming the ships, and the Battle of the Aegates Islands, where Rome's victory was the final tipping point where Hamilcar Barca decided it was time to make peace with Rome. It's an interesting little fact that during this war the Roman population actually dropped, which is contributed to the large losses at sea Rome suffered due to storms and top heavy ships.
After that, Rome could said to anywhere in the Western Mediterranean she liked without meeting much opposition, as there were no other entities that could come near to match her strength. The Roman navy did continue to fight in her wars, even facing Hannibal Barca after he fled Carthage to serve the courts of the East. After this, my knowledge is sparse as it isn't the time frame I've normally studied, or even read of recently. I do know that Pompey cleared the Mediterranean of pirates in a six month time frame, gaining him much popularity.
Sources: The Fall of Carthage, Adrian Goldsworthy In the Name of Rome, Adrian Goldsworthy (Sorry the sources aren't varied for authors, I'm stuck on my ship and don't have access to most of my books)