It was a unique organization within the United States Navy -- it never put to sea, it had no ships, and never had more than about 50 people in its organization.

After the surrender of Germany in May 1945 Tenth Fleet was disbanded in June 1945.

"The Tenth Fleet was formally established on 20 May 1943 and was known as "the fleet without a ship." The Tenth Fleet had its origins in a small unit called the Atlantic Fleet Antisubmarine Warfare Unit which was established in Boston. This small unit reported to the Commander in Chief, Atlantic Fleet. Having been set up on 7 February 1942, its function was to plan anti-U-boat patrol and detection activities on the Atlantic Coast.

During a visit by the Coordinator of Research and Development the availability of scientists at the Massachusetts Institute of Technology working for the National Defense Research Committee was discussed. This was the origin of the Antisubmarine Warfare Operational Research Group (ASWORG) composed of civilian scientist who were instrumental in the technical advances that kept the United States antisubmarine warfare forces ahead of the U-boats.

Shortly after the scientists were brought in, the small Boston unit was transferred to COMINCH headquarters and became the nucleus of the Anti-Submarine Division, which eventually evolved into the Tenth Fleet."

Fleet Admiral King's 3rd and Final report to SecNav in Dec 1945

Here are some internet search results.
The "Tenth Fleet" is referenced in each website linked belew.

In March 1929, eleven years after World War I, the U.S. Congress passed a law authorizing the use of government funds to pay for mothers and widows of fallen veterans to visit their loved ones buried on the battlefields of Europe. This unprecedented program honoring the "Gold Star" mothers and widows was entrusted to the Quartermaster Corps for proper and faithful execution.

At 11:30 a.m. on Friday, 7 February 1930, in the Red Room of the White House, Mrs. Lou Henry Hoover, the President’s wife, reached into a large silver bowl and pulled out the first of 54 unsealed envelopes. Each had in it a card bearing the name of a state or overseas territory. The first state picked was Nebraska -- and the card was instantly handed over to The Quartermaster General.

Two hundred and thirty-one women boarded the Quartermaster steamer S.S. America on 7 May 1930 and left New York harbor for Europe. Over the next three years, ending with the return of the S.S. Washington in August 1933, some 6,693 Gold Star mothers and war widows had made the pilgrimage abroad. No nation before or since has ever so honored the women whose sons and husbands gave their lives in the service of their country.

Image and text above from US ARMY Quartermaster History - Compiled by Dr. Steven Anders, U.S. Army Quartermaster Corps Historian, Fort Lee, Virginia.

AMERICAN GOLD STAR MOTHERS
2128 Leroy Place, NW
Washington, DC 20008
(202) 265-0991

Founded: 1928

Natural mothers whose sons or daughters died in the line of duty in the armed forces during World Wars I and II, the Korean Conflict, the Vietnam hostilities, or in other strategic areas.

Seeks to: inspire patriotism and a sense of individual obligation to the community, state,and nation; assist veterans and their dependents with claims made to the Veterans Administration; perpetuate the memory of individuals who died in our wars; promote peace and good will for the U.S. and all other nations.

Mothers works as volunteers in VA Medical Centers and at Vietnam Veterans Memorial Fund.

Observing GOLD STAR MOTHERS DAY
TITLE 36 -- PATRIOTIC AND NATIONAL OBSERVANCES, CEREMONIES, AND ORGANIZATIONS
Subtitle I--Patriotic and National Observances and Ceremonies
Part A--Observances and Ceremonies
CHAPTER 1--PATRIOTIC AND NATIONAL OBSERVANCES
Sec. 111. Gold Star Mother's Day
(a) Designation. -- The last Sunday in September is Gold Star Mother's Day.
(b) Proclamation.-- The President is requested to issue a proclamation calling on United States Government officials to display the flag of the United States on all Government buildings, and the people of the United States to display the flag and hold appropriate meetings at homes, churches, or other suitable places, on Gold Star Mother's Day as a public expression of the love, sorrow, and reverence of the people for Gold Star Mothers.