I think you'll be fine for your stated use with the RX375; now, if you start turning things up to 11 and ignore distortion, yes, you could damage the speakers with the avr (some avr's come with a feature to limit your volume dial, not sure this is one of them or not, might help if you're not the only one using the avr). Your speakers per that link indicate sensitivity of 89dB, you should be able to reach good volume levels within the capability of the amp in the RX375 in any case. Google "spl calculator" if you want to see the differences in amp wattage and sensitivity can make.
jaygax brings up something you might want to consider for the future, and that's the ability to add outboard amplification; this avr from what I see cannot do that, though, you would need pre-outs for each channel and this apparently only has one pre-out, for a sub.
ps Only the mains are 89dB, the others less but don't think it makes a big difference with your stated use...
"I realize that somebody playing free music isn't as commercial as a hamburger stand. But is it because you can eat a hamburger and hold it in your hand and you can't do that with music? Is it too free to control?" - Don Van Vliet (aka Captain Beefheart) discussing commercial success in the music biz