According to the Quran, can a woman who doesn't wish to get married to stay single? The question applies to a woman who guards her morals successfully, has no physical or emotional desires and wants a peaceful life worshipping Allah.
If yes, are there any ayats in the Quran to state it? And please can you list those ayats?

According to Quran and Hadith, marriage is a very emphasised sunnah of the Prophets (sawa) and Imams (as).

We have many narrations discouraging a person living alone, eating alone and being alone.