That is very bad. Not because of how anti-Semitism is viewed in the west, or because this person is a professor, but because that is a disrespectful thing to say to anyone.
From the career side (which, I don't think is the most important aspect of this), this incident could very well be considered workplace harassment. @Leherenn comments that "This could legally be considered a racist slur in a public setting, which in France is an offense punishable by up to one year of imprisonment and up to a €45k fine." I am not a lawyer and am from the US, so I cannot advise about this aspect, but this is something you should be aware of. Additionally, even if you didn't use a slur, having an emotional outburst in public during a seminar is very inappropriate and will likely make other people question your professionalism. So there could be consequences. Having said that, if you apologize, reflect on what went wrong, and learn the lesson to not use words like this, hopefully you can move forward without permanent damage. But I would not worry about your career for now. You hurt someone, and your first step is to try to make things right.
I would immediately contact the professor and CC the department head and apologize. Be contrite but keep the email brief, something like
Dear Professor X,
I would like to sincerely apologize for my completely inappropriate outburst earlier. The discussion was heated and my emotions got the better of me, and I said a word that was very disrespectful to you and your community that I should not have said, and that I should never say. I will not let it happen again.
You should reflect, and (if you agree) write an apology in your own words. It will mean much more if it is sincerely written and comes from you.
It's better if you get ahead of it (meaning: it is better if you send an apology as soon as possible, before you are asked to send an apology). They may want to take some disciplinary action; be contrite and sincere, and be willing to take the steps they suggest to rectify the situation.
There have been a few comments about also apologizing to the seminar audience and the seminar organizer, so I wanted to add slightly more about this (edit: turned out to be more than "slightly"!)
I think there are several layers to this situation, that need to be unravelled.
- The first layer is that you have hurt a specific person, and you should apologize to that person directly.
- I actually hope that what happens here is that you have a one-on-one in person conversation where the two of you can process what happened together, and that you are able to repair your relationship. But, I think I would start by sending an email. Email gives the person a chance to think before responding, and can hopefully lead to an in-person meeting when they are ready.
- We don't know much about the original incident that prompted you to respond, but I hope this also would be something you discuss one-on-one with the professor.
- I suggested CC-ing the department chair here because this seems like it could be serious enough where you could face disciplinary action, and keeping the chair informed that you are taking steps to rectify the situation could be important later. It was also suggested in the comments to include the organizer of the seminar; that's a good idea too.
- Additionally, I think it is appropriate to apologize the group. I feel that it's probably better to do this after you have apologized to the professor (others may disagree). But... this all should ideally be happening very rapidly so not too much time is passing between steps.
- The main reasons for apologizing to the professor first are (a) the hurt to the professor is larger and more personal and should be dealt with separately, not as part of a group apology, (b) I think it's better to allow them the space to respond to you how they want first in private, because in a group setting they might feel pressure to act a certain way (c) (related to b but more selfish) you don't know how the professor will react, so it's better for you not to be surprised by their reaction to a public setting.
- Part of the group apology should include a statement like "I have apologized in private to the professor" and perhaps "I am taking X concrete steps to atone for my error."
- It's also a good idea to loop in the seminar organizer here. You don't want to catch them off guard by sending a surprise email to the seminar mailing list. They may also be willing to give you a few minutes to say something at the next seminar, if you would want to do that.
- I suspect you will probably be asked to undergo some kind of cultural sensitivity and/or anger management training, and/or have some kind of disciplinary sanction. You should follow these steps and not try to fight them.
- As has been discussed in other answers here, one of the major underlying issues is that you have violated a foundational value in academic culture. It is important to collaborate across cultural lines, because ultimately we value ideas and knowledge, and not tribal differences. In particular, being able to work with people who are not part of your "tribe" is crucial for academic success.
- I also just want to make it crystal clear that I suggested a draft email as a starting off point, but it was not meant to be copied/pasted into an email. The apology needs to be sincere and come from you to be meaningful.
- While bad, I don't think this is fundamentally a hopeless situation. If you are contrite and put in the work to make this right, I am hopeful you will be able to find redemption and move on from this.