The answer is no.  There is no such act or law. However, after the Sullivan brothers were killed, there were several bills introduced in Congress that related to family members serving in a military unit together. None of these bills were enacted into law. However, there are military policies and directives to discourage immediate family members from serving together.

The facts: The five Sullivan brothers of Waterloo, Iowa, enlisted in the Navy on the same day, January 3, 1942. The brothers were assigned to the U.S.S. Juneau, which was torpedoed and sank on November 13, 1942, killing all but 10 crew members.  Although an existing Navy regulation forbade the assignment of the brothers to the same ship, their request to serve together was granted. The existing regulation was issued in July 1942, and was certainly influenced by the loss of the U.S.S. Arizona during Pearl Harbor, with three brothers among the casualties.

Link to the Naval Historical Center for more information.