If the King of Israel sins against another Jew, (e.g cursing, speaking badly, etc.) is he obligated to ask for forgiveness?
While not explicitly addressing your scenario, the Rambam Hilchos Melachim 2:6 writes about how a king should behave and generally conduct himself:
כְּדֶרֶךְ שֶׁחָלַק לוֹ הַכָּתוּב הַכָּבוֹד הַגָּדוֹל. וְחִיֵּב הַכּל בִּכְבוֹדוֹ. כָּךְ צִוָּהוּ לִהְיוֹת לִבּוֹ בְּקִרְבּוֹ שָׁפָל וְחָלָל שֶׁנֶּאֱמַר (תהילים קט כב) "וְלִבִּי חָלַל בְּקִרְבִּי". וְלֹא יִנְהַג גַּסּוּת לֵב בְּיִשְׂרָאֵל יֶתֶר מִדַּאי שֶׁנֶּאֱמַר (דברים יז כ) "לְבִלְתִּי רוּם לְבָבוֹ מֵאֶחָיו". וְיִהְיֶה חוֹנֵן וּמְרַחֵם לִקְטַנִּים וּגְדוֹלִים. וְיֵצֵא וְיָבוֹא בְּחֶפְצֵיהֶם וּבְטוֹבָתָם. וְיָחוּס עַל כְּבוֹד קָטָן שֶׁבַּקְּטַנִּים. וּכְשֶׁמְּדַבֵּר אֶל כָּל הַקָּהָל בִּלְשׁוֹן רַבִּים יְדַבֵּר רַכּוֹת. שֶׁנֶּאֱמַר (דברי הימים א כח ב) "שְׁמָעוּנִי אַחַי וְעַמִּי". וְאוֹמֵר (מלכים א יב ז) "אִם הַיּוֹם תִּהְיֶה עֶבֶד לָעָם הַזֶּה" וְגוֹ'. לְעוֹלָם יִתְנַהֵג בַּעֲנָוָה יְתֵרָה. אֵין לָנוּ גָּדוֹל מִמּשֶׁה רַבֵּנוּ וְהוּא אוֹמֵר (שמות טז ח) "וְנַחְנוּ מָה לֹא עָלֵינוּ תְלֻנֹּתֵיכֶם". וְיִסְבּל טָרְחָם וּמַשָּׂאָם וּתְלוּנוֹתָם וְקִצְפָּם כַּאֲשֶׁר יִשָּׂא הָאוֹמֵן אֶת הַיּוֹנֵק. רוֹעֶה קְרָאוֹ הַכָּתוּב. לִרְעוֹת בְּיַעֲקֹב עַמּוֹ. וְדַרְכּוֹ שֶׁל רוֹעֶה מְפֹרָשׁ בְּקַבָּלָה כְּרֹעֶה עֶדְרוֹ יִרְעֶה בִּזְרֹעוֹ יְקַבֵּץ טְלָאִים וּבְחֵיקוֹ יִשָּׂא וְגוֹ':
In the same way that Scriptures has given him great honor, and everyone is obliged to give him honor, so has he been commanded to be of a subdued and unpretentious heart, as it says, “and my heart is empty within me” (Psalms 109:22). He must not act overly haughty towards Israel, as it says, “lest his heart become haughty over his brothers” (Deut. 17:20). He must be merciful and compassionate to the small and great with regard and deal with their wants and welfare. He must show respect for even the lowest of the low. When he speaks to the entire Congregation together, he is to speak sensitively, as it says, “listen to me my brothers and my people” (I Chronicles 28:2) and “and should you today be a servant to your people…” (I Kings 12:7). He must always behave with extreme modesty, for we have never had anyone greater than Moses Our Teacher who said, “and what are we? your complaint is not against us” (Exodus 16:7-8). He must suffer their burdens and criticisms and anger, “as a nursing father carries the sucking child” (Num. 11:12). Scriptures calls him ‘shepherd’, “To be a shepherd over Jacob His people” (Psalms 78:71), and the (proper) behavior of a shepherd is as has been defined in the Tradition, as it says, “As a shepherd who leads his flock, and with his arm he gathers the lambs, and in his breast he carries….” (Isaiah 40:11).
Being as there doesn't appear to be anything that says he shouldn't not ask for forgiveness, one could argue that asking for forgiveness for a misdeed against another Jew falls exactly in line with the type of behavior a king should exhibit.