Support is fine.
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RishonlyHi X11,
'Share' and 'Support' have different meanings. I think 'Share' might not fit into this context.
RishonlyHi Lazarus,
IMHO, both 'share' and 'support' grammatically work fine;nonetheless,contextually, in the sentence--"I don't share his opinion. I am against what he said"--the first sentence doesn't have a meaningful relationship with with the second sentence. I guess others may have different views on this.