Reuters reports a letter from Captain Brett Crozier, the commanding officer of the USS Theodore Roosevelt, called for the removal of over 4,000 sailors from the ship to place them in isolation.
However Mr Esper told CBS News he didn't think it was time to evacuate the carrier.
"We're moving a lot of supplies and assistance, medical assistance, out to the carrier in Guam. We're providing additional medical personnel as they need it."
Mr Esper says the goal is still to contain the virus and none of the crew were seriously ill.
He admits to not having read the letter from Captain Crozier.
The letter called on navy leadership for stronger measures to save the lives of sailors.
The four-page letter described a bleak situation on board the nuclear-powered, 5,000-person carrier as more and more sailors test positive for the virus.
"We are not at war. Sailors do not need to die. If we do not act now, we are failing to properly take care of our most trusted asset - our sailors," Captain Crozier wrote.
Captain Crozier wrote that the carrier lacked enough quarantine and isolation facilities and warned the current strategy would slow but fail to eradicate the virus.
US officials, speaking on condition of anonymity, told Reuters that nearly 80 people aboard the ship had tested positive for the virus.
However the Navy declined to confirm exactly how many people aboard the Roosevelt had been infected.
Guam cases rise.
Guam, the territory where the USS Theodore Roosevelt is docked, currently has 69 cases of Covid-19.
There was an increase of 11 cases over the past 24 hours.
Two people have died of the coronavirus.
There have been seven recorded recoveries of the disease.