Using Wikipedia's demographic data, here's a list of interesting countries:
1. United States: 34% minority (includes Hispanic whites)
2. New Zealand: 20% (includes Maori, Asian and Pacific Islanders, but not other minorities)
3. Canada: 16% ("visible minorities" only; i.e. non-whites)
4. Australia: 15% (non-European descent)
5. United Kingdom: 14% (only non-whites)
6. Netherlands: 13% (only non-whites)
7. France: 9% (only non-Europeans, but includes self-IDed white North Africans)
8. Germany: 4% (only non-whites)
9. Spain: ~3% (difficult to estimate due to ambiguous status of part-Native Latin Americans, much, much higher if all Latin Americans are considered minority, could be in between if only those self-IDed as non-white are included)
...and New Zealand is the only other western country I could see elecing a non-european PM any time soon. Australia and Canada could probably elect someone of non-typical but still European descent though.
Our Governor General is black, what more do you want?
