Stop Arctic Ocean Drilling Act of 2019 This bill addresses oil and gas leasing in Arctic areas of the Outer Continental Shelf (OCS). The bill declares that it is the policy of the United States that the Arctic Ocean should be managed for the best interests of the people of the United States, including by keeping fossil fuels in the ground to avoid the dangerous impacts of climate change. The Department of the Interior is prohibited from issuing or extending a lease or any other authorization for the exploration, development, or production of oil, natural gas, or any other mineral in Arctic areas of the OCS.