From the short negotiation between Adam and G-d it is clear that Adam is not asking for forgiveness.
So why didn't he ask for forgiveness?
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Bereishis Rabbah 22:13 seems to indicate that Adam didn’t know that he had that option:
פָּגַע בּוֹ אָדָם הָרִאשׁוֹן אָמַר לוֹ מַה נַּעֲשָׂה בְּדִינְךָ, אָמַר לוֹ עָשִׂיתִי תְּשׁוּבָה וְנִתְפַּשַּׁרְתִּי. הִתְחִיל אָדָם הָרִאשׁוֹן מְטַפֵּחַ עַל פָּנָיו, אָמַר, כָּךְ הִיא כֹּחָהּ שֶׁל תְּשׁוּבָה וַאֲנִי לֹא הָיִיתִי יוֹדֵעַ
Adam bumped into [Kayin]. [Adam] said to [Kayin], “What happened with your judgement?” He said to him, “I repented and compromised.” Adam began banging his face; he said, “This is the power of repentance, and I didn’t know!”
Whether this means that he didn’t know repentance existed, or that he knew repentance existed but didn’t know it applied even to a sin like his, I’m not sure. In any event, he didn’t repent because he didn’t think he was able to.