The USC Navy's first major engagment was the Asian-Christian War. They valiantly fought the largest navy in the world in nearly every major waterway. The Asian Navy was also more advanced and more numerous then the Christian Navy who still relied primarilly on 20th Century vessels. They did however destroy them with amazing valiance almost surpassing the valiancy of the Army in the trenches. They had the second highest casualty rate in the war following the USC Army.
The USC Navy had its hands full during the peace time between the Asian-Christian War. They had to deal with pirates, colonization, and protection of merchant vessels. The USC Coast Guard was absorbed into the USC Navy to combine their recources but the Coast Guardsmen still had their own division with-in the USC Navy and they still held their heads high as they replaced the Shore Patrol (Navy's version of the Military Police) and were the only part of the Navy that had authority to perform an arrest.
The USC Navy in the rebellion was feared by all. They became infamous for turning lush planets into nuclear wastelands. They became the hearlder of death and many systems simply surrendered when the first USC Naval Battleships entered. Unfotunantley any system that after their supposed surrender showed any signed of rebelliousness were nuked as the Vatican decided that they were not to risk any-more rebellions.
The USC Navy was for only second time totally and utterly out-gunned. They were killed by the dozens as the ships had nearly no protection against the super-hot plasma bolts burned through their armoured hulls and super heated the air to the point of human combustion. They were also one of the last true defences against the Covenant as the devestating Gravity Missiles were perhaps the only effective means of penetrating the Covenant's shielding. After the Sangheili and Mgalekgolo allied with humanity they often became the commanders of their former enemy's ships. This was the death nail for the Covenant as they were destroyed by the now superior navy.
The USC navy after the war used Covenant designs almost exclusivley as many of the older ships were retired and docked to become 'floating' museums or monuments.
The USC Navy had a significant role in the final destruction of the Covenant at High Charity as they destroyed the majority of the remanant of the Covenant Navy with thoes that they didn't kill escaping with their Kig-Yar masters off into the unknown.
Rebellion of New Quebec
The role of the USC Navy in the conflict was to blockade the system, thats it just blockade it.