Consider the following:
He replied to us even when not solicited/asked.
Can not solicited be replaced with unsolicited?
Solicited can be seen as a verb in the way you want to use it. It’s a non-finite verb, which means it doesn't show tense or person. There is no verb *unsolicit, so there cannot be a verb unsolicited, which must therefore be considered an adjective. By analogy, solicited can also be seen as an adjective as well as a verb.
According to OED unsolicited is an adjective not verb and means not asked for; given or done voluntarily, which could perfectly work in your context.
Yes, you can certainly use "unsolicited" in that way. I'm not sure whether I'd consider it a verb though; I'm trying to imagine how one might unsolicit something or someone.
The use of the word "not solicited" in the original statement is fine.
However, most dictionaries classify "unsolicited" only as an adjective and with the meaning of "not searched or asked for."
So if you want to use "unsolicited," to be safe, I'd suggest:
He gave us a reply even when unsolicited.
He gave us a reply though unsolicited.
It's an example of a past participle ending (-ed) being applied to a noun (not a verb) to create an adjective. E.g. Certificated Bailiff. No such verb as to certificate but it is a perfectly logical way to express in receipt of a certificate.