I believe they are cognate.
Danish dommer is undoubtedly connected to dom "sentence, verdict, judgment, decision" (though I have no page I can point to for proof). Wiktionary says that dom comes from Old Norse dómr "judgment", which is from Proto-Germanic *dōmaz.
And EtymOnline says that English doom is from Proto-Germanic *domaz and that it has cognates including Old Norse domr.
Compare also English deem.