The 67-year-old billionaire was said to have posed for 150 selfies as he toured the vessel, meeting the sailors in a morale-boosting visit to show appreciation for their work.
Nick Wood, executive officer of HMS Ocean, said Sir Richard "had time for a word with everybody" when he toured the ship on Saturday.
The ship dropped anchor off the British Virgin Islands on Friday after powering across the Atlantic Ocean to join the Task Force helping those whose homes and businesses have been devastated.
Laden with more than 60 tonnes of aid, 650 military personnel, landing craft and nine helicopters, its cargo also includes building materials, vehicles and tools.
"He was brilliant, I think he recognises how much the military has done for the islands in particular and what stuck me is how much ownership he feels towards the BVI," said Mr Wood.
"Today was him coming on board pressing the flesh with all the sailors and just saying thank you - posing for 150 selfies - the ship's company was absolutely made up, he loved it.
"It's been a really really good few days for the ship, it's unfortunate that the circumstances are this, but actually it brings out the very best in the sailors that they have gone ashore and they are absolutely exhausted when they get back.
"They just want to have a few hours' sleep and go again. Richard Branson was a lovely thank you to them."
Mr Wood said Sir Richard is a "nice bloke", and that it was a "private visit with no airs or graces".
Hurricane Maria ravaged Puerto Rico and Dominica earlier this week, flooding towns, crushing homes and leaving millions of people without power.
Maria’s death toll across the Caribbean has climbed to at least 25.
Just two weeks beforehand the British Virgin Islands including Sir Richard's private Necker Island were ravaged by Hurricane Irma, one of the most powerful storms in decades.
155mph winds battered homes and businesses, leaving a trail of destruction. Sir Richard said at the time: "Necker and the whole area have been completely and utterly devastated."