ICAO is an umbrella organisation.
You can't get a licence that says just "ICAO".
You get an FAA licence from the US (USA is a signatory to ICAO). If you're in Australia, you get a licence issued by CASA. If you're in Europe you'll get a licence issued by a member state of JAA. JAA is a smaller umbrella but still a part of ICAO. (some countries in Europe aren't a part of JAA - it's complicated
If you want to fly an N-registered aircraft (ie one registered in the US) you'll need an FAA licence. Note, the aircraft doesn't have to physically be in the US, just registered there.
You need to change your licence to fly different country registrations. N-reg is USA, G- reg is UK (but a part of JAA), EI - reg is Republic of Ireland (But a part of JAA), VH- reg is Australian.
The hardest part often isn't changing your licence over (although FAA or CASA to JAA is a challenge!). It is getting the right to work in that country. US require a green card or other visas. I believe there is a visa that will allow people who have learnt to fly in the US to get a 2 year working visa. Note, you must check that one out.