No, it will not damage the speaker. The high tones of a guitar are simply not capable of damaging your bass amp. Even if the amp isn't really suitable for these high tones (and in many cases, they can actually play them just fine), the amp will perfectly survive them. No harm there. The other way around is a much worse idea, because deep bass tones tend to be quite destructive to amps that can't play them!
If you're looking for a decent sound though, you'll notice that a bass amp probably sounds too plain for a guitar. A guitar amp has several effects that make the guitar sound better, which are absent in a bass amp. Unless you're really looking for that specific sound (or going to alter it with floorboards), there isn't really a reason to specifically use a bass amp.