If it's spider mites, yes, they can do lethal damage. Once they get to the webbing stage, they're well on their way.

I assume if you have it in the house it's not gigantic. I would bring it outside and train a strong water spray on the plant, paying particular attention to the undersides of the leaves. Let it dry and then do it again. Let it dry and then, if you want to be very thorough, spray the undersides of the leaves with Safer's insecticidal soap or (what I do and it works well) spray with a mixture of 70 percent isopropyl (rubbing) alcohol diluted with water 1:3 (one part alcohol, 3 parts water, or even 1:2). This should work. But be vigilant.