It is worth noting the Magen Avos commentary on the Mishna where it says:
ולא נתפרש בשום מקום למה נתייחס בשם צדיק יותר משאר התנאים כי כולם היו צדיקים ונראה כי כך היו קוראים אותו כל העם כי בדורו לא היה צדיק יותר ממנו
And it is not explained in any place why he was referred to with the name 'Tzaddik', more so than any of the other Tannaim, because they were all righteous. And it seems that this is how the whole nation would call him because in his generation there was no more righteous than he was.
The Yein Levanon adds to this:
ושמעון הצדיק שמש בכהונה גדולה והיה אחרון לאנשי כנסת הגדולה וממנה יצאה תורה לישראל בדורו
And Shimon HaTzaddik served as the Kohen Gadol and he was the last of the men of the Great Assembly, and from him Torah went to Israel in his generation.
So it would seem that his righteousness stemmed from the fact that he was the main disseminator of Torah in his generation which thereby potentially warranted the title 'Tzaddik'
Indeed, if one looks in Yoma 39a it lists a number of miracles that happened during his tenure as Kohen Gadol which could also perhaps verify why he merited the appellation of tzaddik:
תָּנוּ רַבָּנַן אַרְבָּעִים שָׁנָה שֶׁשִּׁמֵּשׁ שִׁמְעוֹן הַצַּדִּיק הָיָה גּוֹרָל עוֹלֶה בְּיָמִין מִכָּאן וְאֵילָךְ פְּעָמִים עוֹלֶה בְּיָמִין פְּעָמִים עוֹלֶה בִּשְׂמֹאל וְהָיָה לָשׁוֹן שֶׁל זְהוֹרִית מַלְבִּין מִכָּאן וְאֵילָךְ פְּעָמִים מַלְבִּין פְּעָמִים אֵינוֹ מַלְבִּין וְהָיָה נֵר מַעֲרָבִי דּוֹלֵק מִכָּאן וְאֵילָךְ פְּעָמִים דּוֹלֵק פְּעָמִים כָּבֶה וְהָיָה אֵשׁ שֶׁל מַעֲרָכָה מִתְגַּבֵּר וְלֹא הָיוּ כֹּהֲנִים צְרִיכִין לְהָבִיא עֵצִים לְמַעֲרָכָה חוּץ מִשְּׁנֵי גְּזִירֵי עֵצִים כְּדֵי לְקַיֵּים מִצְוַת עֵצִים מִכָּאן וְאֵילָךְ פְּעָמִים מִתְגַּבֵּר פְּעָמִים אֵין מִתְגַּבֵּר וְלֹא הָיוּ כֹּהֲנִים נִמְנָעִין מִלְּהָבִיא עֵצִים לַמַּעֲרָכָה כׇּל הַיּוֹם כּוּלּוֹ
§ The Sages taught: During all forty years that Shimon HaTzaddik served as High Priest, the lot for God arose in the right hand. From then onward, sometimes it arose in the right hand and sometimes it arose in the left hand. Furthermore, during his tenure as High Priest, the strip of crimson wool that was tied to the head of the goat that was sent to Azazel turned white, indicating that the sins of the people had been forgiven, as it is written: “Though your sins be as crimson, they shall be white as snow” (Isaiah 1:18). From then onward, it sometimes turned white and sometimes it did not turn white. Furthermore, the western lamp of the candelabrum would burn continuously as a sign that God’s presence rested upon the nation. From then onward, it sometimes burned and sometimes it went out. And during the tenure of Shimon HaTzaddik, the fire on the arrangement of wood on the altar kept going strongly, perpetually by itself, such that the priests did not need to bring additional wood to the arrangement on a daily basis, except for the two logs that were brought in order to fulfill the mitzva of placing wood upon the arrangement. From then onward, the fire sometimes kept going strongly and sometimes it did not, and so the priests could not avoid bringing wood to the arrangement throughout the entire day. (Sefaria translation & notation)
As far as your point - 'Was he called so by the Greeks (Alexander the Great)?', Rabbeinu Yonah definitely points to the high esteem he was shown by Alexander the Great. It writes there:
שמעון הצדיק היה משיירי כנסת הגדולה. והיה כהן גדול כמו שאמרו במסכת תמיד (כ"א) שהיה יוצא מלובש בבגדי כהונה כנגד אלכסנדרוס וירד והשתחוה לו. אמרו לו עבדיו אדוננו מלך כמוך משתחוה ליהודי זה. אמר להם דמות דיוקנו של זה אני רואה במלחמה ונוצח:
Shimon the Righteous was from the remnants of the Great Assembly. And he was a high priest, as they said in Tractate Tamid 21 (Yoma 69a) that he had gone out dressed with the priestly garments in front of Alexander [the Great] and [Alexander] went down and bowed before him. His servants said to him, "Our master, A king like you should bow down to this Jew?" He said [back] to them, "I see the image of this one in war and I am victorious."