ICAO do not issue licences but they make recommendations that are adopted by its member states, so an FAA or EASA licence will also be ICAO compliant, as will be yours, but there is still a world of difference, as the ICAO standard will be a minimum.
Even within EASA there is a difference between countries!
If your licence is a PPL, you will find that most countries accept it without question, but if it is a CPL or ATPL, you might find it best to convert it - as mentioned, FAA or EASA seem to be the best choices. An FAA licence can be converted directly to a Canadian one, and the Australians appear to accept an EASA one by taking a law exam, and that can be directly changed for a New Zealand one. If ICAO did issue licences, life would be a lot easier
