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Judo might be an option, it is a standardized “sport” martial art based on jujutsu. The sport of judo does not allow striking techniques, it is focused on throwing (when in a standing position) and holding, choking and joint locks (elbow only) in the prone position. Compared to western sports, I think of Karate as similar to Boxing and Judo as similar to Wrestling.
In order to be successful, you do need to be aggressive, it is a form of combat, and people do occasionally get injured, but you’re not going to be punching someone in the face.
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While there is no striking in judo and it is an extremely good workout, it’s pretty much necessary to work with a resisting opponent - randori (matwork) is free grappling with another person. Throws are learned by throwing people, and by being thrown. It’s a sport and it’s fun but it’s certainly as “aggressive” as sparring would be (albeit without punching/kicking).
Not all styles of jiu-jitsu and aikido are totally “peaceful” either, definitely check with some local instructors. While they may not focus on strikes, all of the joint locks, throws and chokes are offensive in nature and are best practiced with other students. When I was doing budo aikido/goshinjitsu in college we did plenty of sparring and wrestling.
The things that came to my mind when I read the OP have pretty much been mentioned - tai chi and yoga, or take some cardio kickboxing/boxing classes (many gyms offer those, the goal is aerobic exercise without actually sparring). Like I said, you can also check around your local martial arts schools for “soft” styles, or styles where you focus more on forms (kata).
AFG, do you mind if I ask whether it’s specifically punching/kicking someone (even in a sparring setting, with pads) that you don’t like, or is it anything where you are working with a resisting opponent? If it’s the former, and wrestling doesn’t bother you, then you have many more options (judo and jiujitsu for example). If it’s the general agressiveness some of that may be the tone of the instructor - I’m sure that some places have a real “macho ass kicker” vibe and others go for a more fun attitude (while still being vigorous about the art itself). I’ve seen schools that have women-only classes as well if that’s an issue.