While most European doctoral courses require its applicants to hold master's degree, many doctoral courses in US often do not. In these graduate schools, by satisfying conditions set by the schools, students advance to candidacy, when the school may or may not grant them master's degree.
Suppose the school does not. Then, can a Ph.D. candidate at the school be admitted to a European doctoral course, which usually require its applicants to have master's degree?
Although the term "PhD candidate" is used in both continents, a Ph.D program in the USA comprises of: Master (2 years) + Doctorate (3 years), meaning it takes at least 5 years, whereas in Europe comprises of just the latter: Doctorate (3 years). This means that yes, a Master's degree or something deemed equivalent or superior is required to enter the doctoral course in Europe. The fact that it's "not required" in the US is an illusion, it's just because the Master's is (sort of) included IN the Ph. D program. In the end you will get your Ph. D degree at around the same age than students in Europe.
If your University in the USA does not deliver a Master's degree per se, you will have to check with the prospective university in Europe to see whether they will count your research experience (preferably 1-2 years of lab research after your 4-year Bachelor's, which you could sum up in something that looks like a Master's thesis) as being equivalent or superior to a Master's degree. I believe it is case by case.
Also, please keep in mind that some countries like France require you to have a scholarship or to secure some kind of funding prior to admission to cover your living expenses for the entirety of the Doctorate (3 years) since Ph.D candidates have a status similar to paid staff.
The UK is probably one of the only countries in the world where it's OK to do just a Doctorate (3 years) right after your Bachelor's (=if you compare to other countries, you could say it means skipping the 2 years of Master's), however, I don't think this is highly valued on the international market, unless you managed to end up with the same amount of work and publications than the people who did Master (2 years) + Doctorate (3 years).
//Speaking for science majors//
One should note that the reason why there's such a big difference between the US and the European systems is the way master's degrees are handled. In the US, the master's degree is viewed as a "graduate" degree, to be completed after the bachelor's study, which is effectively independent. In Europe, however, the bachelor's degree is largely considered a precursor to the master's degree, which is normally done in succession after the bachelor's degree is complete. Doctoral candidates (post-master's) in Europe do not normally have significant coursework requirements associated with the degree.
Consequently, for most PhD positions in Europe, a master's degree is considered a prerequisite. The exception to this are programs that are more or less patterned after the US model, in which a master's degree is acquired as part of a larger PhD program. However, even in these programs, the acquisition of the master's degree is not merely "an option," as it is in many US programs. Instead, it is a necessary component of what amounts to a dual-degree program.
