Libby Health Care Act - Amends federal bankruptcy law to prohibit the court from entering an order confirming a plan of reorganization under chapter 11 involving a responsible party, or issuing an injunction in connection with such order, unless the responsible party: (1) has established a health care trust fund for the benefit of individuals suffering from an asbestos-related disease or illness; and (2) has deposited not less than $250 million into such health care trust fund.

Defines "responsible party" as a corporation: (1) that has engaged in mining vermiculite that was contaminated by tremolite asbestos; (2) whose officers or directors have been indicted for knowingly releasing asbestos into the ambient air and knowingly endangering the residents of Libby, Montana, and the surrounding communities; and (3) for which the Department of Justice has intervened in a bankruptcy proceeding.

Requires any payment received by the United States for recovery of costs associated with the actions to address asbestos contamination in Libby, Montana, to be deposited into such fund.

Sets eligibility criteria for medical benefit payments from the fund.