Consider the following:
He replied to us even when not solicited/asked.
Can not solicited be replaced with unsolicited?
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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.
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.