In a brief statement released Saturday afternoon, the Cleveland Clinic leader said he was "humbled and honored" to have been considered.
BUT, he wrote, his commitment remains with the Clinic and its patients.
Cosgrove, who turns 74 in July, cited unfinished business at the non-profit hospital system.
The Clinic has grown to a global brand since Cosgrove took the helm in 2004. It has about 40,000 employees and serves millions of patients each year. In 2013, it reported $6.5 billion in revenue.
Major projects currently under way include the construction of a hospital in Abu Dhabi and the expansion of its medical school.
The White House did not release a statement Saturday. Cosgrove wrote that the decision was "extraordinarily difficult."
In his statement, Cosgrove noted that he is a former heart surgeon and veteran, adding "I have great respect for the care provided to the Veteran community and for those who work to care for them."