I had one bad timing belt tensioner, so I got a replacement. The spring was also over stretched and so I was expecting a new spring to come with the new tensioner. Well, it didn't. How critical is it that I replace the spring with another with the same damping constant? I guess what I am asking is can I just get one from the hardware store that has about the same dimensions?

the spring is there tp oretty much take the slack out of the belt as you install it. once you tighten the tensioner bolts the spring is just there, the tension is on the bolts. tou can push down on the tensioner pulley the same as you tighten it as well

As MilesFox noted they are there to assist you when installing a belt. The springs on mine are shot. One is complete junk (stretched otu like you described) and the other is pretty pointless. I threw the one away and left the other on there. Works just fine but makes installing the belt and getting it tight more tricky.