In backyard football, if a player injures another during a tackle, is he legally responsible to pay medical expenses?

Potentially yes, the owner of the property might also share in the liability. However, the person who was injured, presumably, entered the game with the knowledge that he or she might be injured while playing. A court would determine the amount of liability on a percentage; i.e. the player who was tackled might pay 33%, the property owner 33%, and the player who was injured would be responsible for the remainder.

LOL. In Canada you just go to the hospital and there are no medical bills.

WHAT A COUNTRY.

Not completely alone. A person injured can sue the person who hurt them, and the property owner. The property owner's insurance company might be inclined to settle. The person who tackled someone should really get a lawyer because they need legal defense, and someone to negotiate on their behalf. Some expenses might be covered by the injured person's insurance. However, there is often a steep out of pocket deductible first before coverage is made.