In a way that strongly suggests guilt or wrongdoing; in a manner that condemns or incriminates.
From the adjective damning (present participle of damn used as an adjective) plus the adverbial suffix -ly. The meaning evolved from 'in a manner worthy of damnation' to 'in a manner that condemns or proves guilt.'
When a lawyer says 'damningly' in court, it's a word that carries echoes of religious judgment — we're essentially calling something 'worthy of eternal punishment' when we just mean it looks really guilty.
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