WASHINGTON—A document compiled by Democrats on a Senate committee levels new allegations against Ronny Jackson, President Donald Trump’s embattled nominee to head the Department of Veterans Affairs, who said Wednesday he will keep fighting for the job despite a growing list of accusations including charges of reckless drug-dispensing practices.

Dr. Jackson, speaking briefly Wednesday with reporters at the White House, rejected one of the most serious allegations contained in the report—that he wrecked a government vehicle...