I think all hospital and nursing homes use large washing machines and very hot water. Sanitation is a huge issue in those places.
Quilts are inherently somewhat fire retardant in that they will not burst into flames the way thin draperies will. The thickness of the layers of a quilt slow down the fire. This is because fire is fueled by oxygen. A thin drapery lit from below will burst into flames because air can easily reach both sides of the fabric. A quilt made with cotton or wool batting will burn very slowly because the ratio of air to fabric is much smaller, and the burned portions will turn to ash. A quilt made with polyester batting will also burn very slowly for the same reason, but the burned portions will consist largely of a gooey substance that can stick to skin and create third degree burns.
Fire retardancy is quite a complex issue. It seems to me the hospital person who talked to the OP does not really understand what "fire proof" means. I sincerely doubt that all of the linens and blankets used in a hospital are "fire proof".