Is it possible to pursue a research career inside a university without teaching duties?
I don't like to teach, I love research and I love being affiliated to a university.
The universities I am interested are based in Europe, UK and USA.
Yes. In the UK such positions are called Research Assistants or Research Associates. There are also Postdoctoral Researchers.
These are full-time salaried researchers who are employed on specific grant funded projects. The only down-side is that they are often fixed-term contracts linked to the fixed term research grant; however, UK employment law does offer some protection, because if you land several sequential fixed term positions at the same institution you are regarded as a permanent employee with greater rights when the funding ceases.
Such positions are advertised in the Academic press (such as the Times Higher Education Supplement) or on jobs.ac.uk.
Some universities in the US have Research Professor positions available. These usually are non-tenured positions, and they often don't have any salary lines associated with them; the research professor's salary comes entirely from the research grants that the research professor brings in as a Principal (or Co-Principal) Investigator, or participates in as a Senior Investigator (someone else, possibly even from another university, is the Principal Investigator) on the research grant or contract. There are similar positions at the associate and assistant professor levels. Such positions are often created at the behest of the applicant: a hot-shot researcher with no interest whatsoever (and possibly no experience too) in teaching wants to bring his lab/research program from, say, industry or a free-standing research institute, to a university and offers to do so if the university creates a position from him.
In France the CNRS offers each year a few research-positions in a considerable range of disciplines. (The initial hiring is usually done at a level comparable to assistant professor.)
The recipient of such a positions is then typically assigned to some university where they will work in the laboratory corresponding to their field.
These are permanent positions, and there is an option for later promotions.
For certain disciplines there are other organization that offer similar types of jobs (such as Inria, for computer science).
With such a position it is also possible to supervise students, and it is often also possible to teach a bit on the side (if at some point, one wants to).
Thus, I would say, yes, this is possible. But, as said in another answer, there is a lot of competition.
There are quite a number of fixed-term positions (research associate or postdoctoral research assistent), funded on research grants, typically for 2-3 years and advertised worldwide. There are also research fellowship with various funding agencies, typically for 3-5 years (or even longer) for early-career researchers.
However, when it comes to permanent positions, the options are limited. There are some places where you can avoid teaching undergraduates. However, any (good) researcher should (and will want to) have postgraduate students (e.g. PhD).
Some Universities have research-only positions and there are also research institutes, for example the German Max-Planck institutes, where researcher still have access to postgraduate students.
However, having said all that I must warn you.
It's perfectly possible. I'm at a UK Russell group university. I work with two professors with no teaching commitment whatsoever.
(This isn't the 'research professor' post described in another answer. These are people with the same tenure and status as professors with a more diverse portfolio - they just don't have any teaching commitments)
The university has a whole pathway from PhD through to professorial level for people with no teaching commitment (and similar pathways for people with no research commitment, and for people with a mix of teaching and research).
Until a couple of years ago I was Director of a department which employed multiple staff at all these levels as research only, with no teaching commitment.
It seems unlikely that my university is the only one that does this in the U.K.
I can imagine this answer being just the opposite of what you're asking for, but can I ask you why you don't want to teach?
I remember that, when I was in my twenties, I didn't like teaching as well: I was so nervous while standing before a group of people that I froze and hence, I didn't see myself as being capable of teaching. So, when I graduated from university I started looking for a job in the industry but I didn't find anything right-away.
As a result, I started teaching anyway, I was standing before some classes who didn't care at all about what I was saying ("Sir, why do we need mathematics for stirring in the soup?"), so after some months I was happy finding another job in the industry, where I'm working ever since.
Now however I'm still enjoying the results of what I learnt at that school: I can still very easily stand before groups of people and say something without freezing, I even did the presentation of a quiz during the wedding of one of my best friends, in front of 150 people. This would not have been possible without that teaching experience.
I'm not simply writing this for explaining my own personal situation: I am aware that public speech is a big burden for lots of people, and the more those people are interested in investigation and research (like you), the more they tend to have difficulties with this matter. (in case this does not apply to you, feel free to delete this answer)
One other thing: as mentioned before indeed much universities give the opportunity to do research without teaching duties, but on the other hand, if you present yourself at a university, refusing one of the basic task of a university, you might make a very bad first impression.
Therefore, I'd propose you to present yourself at a university, being very willing to do research, but you mention that teaching might be problematic, and you mention the specific reason(s). As universities are very good at problem solving, in case they need you to fulfill teaching duties, they will make sure you get the proper guidance and you might very well end up stronger than you imagined at first.
Firstly, it is quite simply incorrect to say that research institutions are void of prestige, especially amongst the academic community, whose respect is presumably what you are trying to garner. I would also suggest that a long-term career in research - with all of the ups and downs, the long projects, setbacks, funding rejections, problematic reviewers and more - is only sustainable if your primary motivation is to learn more about your field and contribute to its development, rather than prestige.
Secondly, your comment about postdocs indicates that you do not understand how academic progression works. It is normal to carry out 2 or (usually) more postdocs before finding a tenured, permanent or long term post somewhere. Postdocs are an invaluable experience, where you learn about your subject, how to carry our research, the different cultures and working practices of other institutions and other counties etc. Indeed, many researchers find they were more productive during their postdocs than during their fixed posts, which come with additional admin, teaching and/or outreach duties. It would be foolish to suppose you can jump straight into a permanent post without gaining this insight into academic and building up your portfolio of articles, conference attendance and seminar practice.
Thirdly, how did you get to where you are? Do you not think you have a duty to give back to the academic community who trained you, rather than rejecting the notion of sharing your knowledge with students, as your professors did with you?