My biggest concern would be that you shredded the mate. My philosphy is that if one goes (and it's not obviously road debris) the other is about to go. Add to that the fact that these sound like the stock tires which are not always the best of quality or strength and it's just another nail in the coffin so to speek. It also sounds like the "old" tire is up front (?)...you REALLY REALLY do not want to have a catestrophic failure on the front wheel, so *I* would replace it or at least move it to the back (more trouble than it's worth) ASAP.

But to answer your question, yes it's fine for a few rides.