I admit I should've worded my post differently. I should've said both countries viewed their radical neighbors with suspicion, i.e. regions that are de facto controlled by radical Islamists.
Yes, India and the GCC have many centuries of historical cultural and economic ties. And even before the modern states were established, Indian civilization has had a huge influence on the cultures and polities of the Persian Gulf region and the Arabian Peninsula for thousands of years. Elamite civilization, Dilmun civilization, etc.
I think the future's bright, and I think even China will have fruitful ties with India and the progressive countries of the Arab World (e.g. the UAE) over the coming decades.
Really, there's nothing to worry about. There's not going to be a war between India and China. This century is Asia's century, and it belongs to both countries.