Can an american dentist practice in europe?

No. Most of the developed countries work on mutual agreement of recognition of their degrees. Most countries require a mutual agreement. Australia and UK recognize their degrees mutually, so they can work in both places. UK has an exam called IQE, it is 3 part exam. If you pass this exam you will get full dental license. United States degrees are not recognized in UK . But you could get a temporary license to work in UK. All you need to get this temporary license is, your dental school should be listed in the World Heath Organization. But the catch is you will get this license ONLY after you find a job, and nobody wants to give you a job unless if you have full UK license. If you have American degree you will have higher chances of getting a job. you can work only for specified time.