I sincerely apologize for the inappropriate remarks made earlier.

Literal

Earlier inappropriate remarks [subject-が] existed matter [object-を] deeply apologize do [humble-いたします].

不適切な発言 = 'inappropriate remarks.' 深くお詫び致します is a very formal apology using humble お詫び + 致す (humble する). ありました事 nominalizes the event. Standard public-apology language.