Yes, you need to replace the crank arm, if tightening the bolt does not make it 100% solid again. No doubt the crank arm had been loose for days, and had you tightened it earlier you might have "saved" it.
And there's some danger that you have damaged the crank axle as well, meaning the bottom bracket cartridge will need changing out as well.
I have speculated that a slightly misshapen crank could be saved by fashioning a shim of aluminum sheet metal (ie, beer/pop can), but I've never had need to try that, and it sounds like yours is in too bad of shape for that to have a chance to work.
(I'll add that replacing just the arm vs a larger assembly is a matter of feasibility and economics. If you can't find a compatible replacement arm you may need to just install a new crankset, or you may discover that the entire crankset is cheaper than the replacement arm.)
As others have pointed out below, the OP's description is more consistent with a failure at the joint between the crank arm and the pedal than between the crank arm and the crank shaft. If this is the case there is some chance that a "Helicoil" can be installed to fix it. This can be done by a skilled bike mechanic or a machine shop.
(In this case the pedal is likely toast, however.)