Roundly means firmly and thoroughly, said of criticism. Roundly condemned, roundly beaten, roundly rejected.
It attaches to a short list of verbs — condemned, criticised, defeated — and always intensifies. The round in it means complete or full, the same sense as in a round dozen, rather than anything about shape.
1adverbroundlyFirmly and thoroughly.
“The proposal was roundly rejected by every member of the committee.”
roundly is a degree.
Round plus -ly, from round in the sense of full or complete, as in a round number.
Middle English
roundly
English, today
roundly
From Middle English roundly, roundely, roundliche, equivalent to round + ly.
Etymology from Wiktionary, CC BY-SA