Probably not safely. Cropping an image to just the head and retouching it still creates a derivative use of someone else’s copyrighted photo, and the community answers generally do not consider that clearly protected by fair use. Fair use is highly fact-specific and often unclear unless tested in court, so no one can guarantee it.

While a small, noncommercial avatar is unlikely to attract legal attention in practice, that does not make it lawful. Using only part of the image or altering it slightly does not automatically make it fair use.

The safest approach is to use a photo that is public domain, or one released under a license that allows this kind of reuse and modification (for example, some Creative Commons licenses), and follow any attribution or share-alike requirements. Otherwise, get permission from the copyright holder.

So: it may be arguable in some cases, but it is not something you should assume is fair use just because it is small, personal, or edited.