And one of His [God's] signs is the creation of the heavens and the earth and the diversity of your tongues and colors; most surely there are signs in this for the learned.
As the context makes clear, this is part of a more general Qur'anic pattern in which this universe itself - the normal, everyday events that we look at as just the way things are - is identified as a sign from God; the Creator's nature is reflected in His creation. So the thrust of the verse is that linguistic diversity is a part of nature, and as such a part of God's plan for the world.
Another relevant verse, which, in light of 10:47 ("for every nation there is a messenger"), puts the idea that the Quran being written in Arabic makes Arabic the best of all languages into perspective, is 14:4:
And We [God] did not send any messenger but with the language of his people, so that he might explain to them clearly; then God makes whom He pleases err and He guides whom He pleases and He is the Mighty, the Wise.