Open Skies Treaty Stability Act This bill prohibits the President from suspending, terminating, or withdrawing the United States from the Open Skies Treaty, unless the Department of Defense and the Department of State jointly certify to Congress that (1) Russia is in material breach of the treaty and is not taking steps to return to compliance, (2) withdrawal from the treaty is in the best interests of U.S. national security, (3) there is a comprehensive strategy to mitigate against reduced military transparency, and (4) other countries that are party to the treaty have been consulted as to Russia's breach and U.S. intentions to withdraw. (The treaty allows participating countries to fly unarmed observation aircraft over each other's territory and places various limits and requirements on such flights.)