Born in New York City, New York, Acosta after serving in the United States Navy from 1943 to 1946, during World War II, and participated in the D-Day Invasion of Normandy (Utah Beach) attended Rutgers School of Law – Newark where received a Juris Doctor in 1951.

He was in private practice of law in Hackensack, New Jersey, from 1953 to 1954, and was then a special agent at the Federal Bureau of Investigation field office in San Diego, California, Washington, D.C., and Miami, Florida, from 1954 to 1958.

[1] In 1980 President Jimmy Carter named him United States Attorney for the District of Puerto Rico, serving until 1982.

[3][1] Acosta was perhaps best known for presiding over the complex litigation resulting from the Dupont Plaza Hotel arson, a disaster that claimed 98 lives.

His "masterful and innovative handling of the complex mass disaster litigation" led to the approval of a $220 million settlement of the lawsuit which had involved 2,400 plaintiffs, 250 defendants, and three trial phases over 19 months.