The Torah introduces the command of Birkat Ha’mazon with the verse, "Ve’achalta Ve’sabata U’berachta" – "You will eat, you will be satiated, and you shall bless" (Debarim 8:10). The Sages understood the word "Ve’sabata" in this verse as referring to drinking – specifically, to drinking wine. On this basis, they established that even one who drank wine during his meal, and is in a state of inebriation, is required to recite Birkat Ha’mazon. Since the Torah speaks in this context of somebody who drank wine, we may deduce that such a person is required to recite Birkat Ha’mazon despite his state of intoxication. When it comes to prayer, one who is inebriated to the point where it would be disrespectful for him to speak to a king, because he cannot pronounce his words clearly, may not pray. With regard to Birkat Ha’mazon, however, the Torah allows – and requires – one to recite this blessing even in a state of inebriation, as long as he is able to recite the words.
Summary: One who drank wine during his meal and is inebriated must nevertheless recite Birkat Ha’mazon, as long as he can pronounce the words, even if he cannot speak as clearly as usual.