What percentage of the world's airline operations are in the USA? And how does this number compare to other first world countries such as the UK and Canada? How about places like China and India? And finally smaller third-world countries?
It may be a bit hard to find this data for many different countries, especially if you want to separate takeoffs and landings.
According to this data for the US, there were 757.1 thousand flights in May 2014, including both domestic and international. Of those, 683.0 thousand were domestic, and 74.1 thousand were international.
Lots of information is available here. This indicates 8.3 million domestic and 1.4 million international flights in the US during 2013. This includes all carriers an all airports, including domestic all-cargo flights.
The best data for Europe I could find is only in passenger count (the more common statistic). It looks like you have to request access to get more detailed information.
However, this collection of data from Eurocontrol showed 9.548 million "IFR Flight Movements" in 2012. This would include all IFR flights, regardless of their type. A graph at the bottom shows figures for 2008. This shows a little over 8 million internal, and a little under 2 million arriving/departing. Since a domestic flight would be two movements (departure and arrival), this is 4 million and 2 million flights. As expected, Europe has a much higher percentage of international flights (about 1:2 international in Europe, versus about 1:9 in the US).
Data from the UK show that there are about 800000 domestic and 1000000 international flights per year.
You could also order a year of World Air Transport Statistics for only $799.
tl;dr: Somewhere between 28.3% and 37.8%.
U.S.-flagged air carriers operating in the U.S. account for 28.3% worldwide scheduled operations while all air carrier traffic in North America accounts for 37.8% of worldwide traffic. Thus, the sum of all scheduled passenger operations in the U.S. is somewhere between those two figures.
Of the airports that participate in the Airports Council International statistics, 37.8% of aircraft movements for the year ending May 2014 were in the North America region, which does not include flights in the Latin America/Caribbean region. It appears that Mexico is included in the Latin America/Caribbean region, so it can be safely assumed that the vast majority, though not all, of the North American region traffic is in the U.S. Canada, of course, is also included in the North American region. The report linked above lists 24,571,773 movements in the North American region out of 64,949,170 for all reporting member airports worldwide.
The U.S. Department of Transportation maintains official statistics for U.S. air carriers. According to the U.S. DoT's Bureau of Transportation Statistics, in the year ending June 2014, there were 9,177,347 departures on U.S. airline passenger flights. Note that 'movements' in the ACI statistics above include both departures and arrivals as separate movements, so for purposes of comparison, the 9.18 million departures would equal 18.35 million movements. In order to get information for the exact same period measured by the ACI report (June 2013-May 2014 vs. the July 2013-June 2014 data in the DoT report,) we can subtract June 2014 departures and add June 2013 departures to the U.S. data. These numbers are given on the same DoT page linked above as 791,896 for June 2014 and 810,721 for June 2013. This yields 9,196,172 departures or 18.39 million movements, which accounts for 28.3% of ACI's worldwide movements for that period.
Note that flights on non-U.S.-based airlines arriving in or departing from the U.S. are not included in the U.S. DoT figures, nor are cargo flights or non-revenue flights (such as repositioning flights) included in their figures, so the actual percentage for the U.S. would be somewhat higher than that 28.3% figure, but lower than the 37.8% figure for all of North America.
Note also that the above links are unfortunately not permanently linked to the time series mentioned here. They will both continually be updated with the most up-to-date information, so, if you're viewing this answer significantly after it was posted, the exact figures given in those links will be newer than what I have included in this answer.
You can get a good feel for the numbers from this world map which shows total passengers carried around the world. Note: click on the "Points" option at the top left of the map to show the figures labelled on the map. US dominates when it comes to air travel, when domestic and international are included, with 743 million passengers in 2013. China came second with 353 million, then UK with 118 million.