Nope.I don't think you're likely to find such a thing. That would be a disaster for staffing, as it takes officers off the street to ride ambulances to the hospital. There are a few police-based EMS systems in my area of NJ, but they all use Community Service Officers to work as EMTs.
I suspect the only full-time Medic/Police combo you'd see would be on SWAT teams and similar groups. But that's just my suspicion, I could be wrong.
One of the best known LEO/Medic programs is the Maryland State Police Aviation Unit that works out of Shock/Trauma at U of M. As an aside, the full name of Maryland's Shock Trauma is the R. Adams Cowley Shock Trauma Center, named for the father of emergency medicine and the originator of the "Golden Hour" concept.
To become a Public Safety Officer (PSO) requires a level of dedication to success not required of most careers. A Public Safety Officer performs the role of Police Officer, Firefighter and Emergency Medical Technician (EMT), something accomplished by separate departments in most municipalities. The role of a Police Officer in society is not to be taken lightly, it is a personal commitment that one must make to live a life that is beyond reproach. A Firefighters job requires a high level of bravery and skill to accomplish a dangerous mission. And EMT's need the motivation and interest to care for people in distress during their most difficult moment in life. The job is a serious one, and requires a level of dedication that is sustained throughout your career.
I still remember the first time I saw a bunch of police cars skidding to a stop outside a building where the fire alarm was going off, and watching them open the trunks on their Crown Vics, and pull out their turnouts!
Nassau Long island: http://www.police.co.nassau.ny.us/eab.htm
The town of Steilacoom Washington, south of Tacoma, uses PSO (Public Safety Officers) http://www.townofsteilacoom.com/town_offices/public_safety.htm