Larry Kelly faces several charges, including felony assault.
Authorities say the 36-year-old was in the ER just after 8 a.m. on Saturday when he became agitated and ran outside.
That's when he allegedly jumped into an idling SUV and struck a ChristianaCare constable, driving over him.
Kelly then reportedly fled on Churchmans Road.
A short time later, troopers say they tracked the stolen SUV to an area near East Newport Pike and Forest Drive, where it had been abandoned.
Investigators say Kelly fled from the car and foot, and he was later found attempting to enter a home on Maple Road.
It was discovered that Kelly did not have a valid driver's license, according to police.
The constable was treated at the hospital for serious injuries.