The justification for proselytizing in Christianity at least is, "Do unto others as you would have them do unto you." That is the essence of your question. If someone doesn't have the truth and you love them, you share it with them. They try to convert others to their faith. They call it Witnessing.
The Jewish way is learned from Hillel HaZaken from Shabbat 31a. Interestingly enough, the conversation is with a non-Jew who comes to him to convert if Hillel can teach the whole Torah to him while the non-Jew is standing on one foot.
Hillel's response was:
דעלך סני לחברך לא תעביד זו היא כל התורה כולה ואידך פירושה הוא זיל גמור.
"What is hateful to you, do not do to your neighbor, that is the
entire Torah, the rest is just commentary, now go and study."
We do not seek to convert everyone to our faith, just as we do not desire others to attempt to convert us from our faith. The Torah values diversity. This follows an explanation of the Lubavitcher Rebbe to comments from Rashi of Bereshit 30:35-37.
Rashi’s comment that the streaks (b’rudim) were white can be understood in reference to an earlier verse. When Ya’akov peeled stripes from the wooden sticks (30:37) he “uncovered the white layer underneath.” When the animals saw these sticks and mated, the offsprings had markings instead of uniform coloring like their parents -similar to the sticks which also had various colors, the color of the bark and the revealed white of the fresh wood underneath. Rashi therefore derives that the streaks were white, since they resulted from the animals’ gazing at the sticks, which also contained white (cf. 30:35 and Rashi there).
We have often discussed the fact that Rashi’s commentary, although it
primarily addresses the simple meaning of the verse, alludes to deep,
esoteric concepts in Torah.
As previously mentioned, “b’rudim” represents the world of “Tikkun,”
which is characterized by the presence of differing entities which
nevertheless coexist and intertwine together. This aspect is compared
to a multi-colored object, the beauty of which derives from the
combination and proximity of the many colors.
This is like is found in regard to the coat of Yosef (Bereshit 37:3). What distinguished his coat was that it contained many colors.
And so, in keeping with that idea, Jews do not seek to convert non-Jews to Judaism. Like with any healthy ecosystem, our diversity is the way of life.