Oppose itself means talking, reacting or in general, being against something. Hence it makes no sense at all to use the word against here again. Also the adverb strongly should be placed before verb oppose to make sense.
He strongly opposed my suggestion. He refused to discuss it.
React would be correct here, however the adverb placement would matter.
He strongly reacted against my suggestion. He refused to discuss it.
Disagree is just not correct and doesn't make much sense in this context. If you wish to use it, then:
He strongly disagreed with my suggestion. He refused to discuss it.