Does someone need to physically die?
No. This is going to depend on how you choose to write the story.
Most villains are written in a way that their death is the only way to prevent them from doing any more harm to innocents. When that is true, we readers expect the hero to not be a coward, and kill the villain, because it has been made clear to us that doing anything else means letting this predator murder again, or rape again, or enslave again, or a perpetuation of whatever harm the villain is doing.
Although personal moral calculus may vary, most people subscribe to the notion that your first time is always the most trepidatious, but the more you do something the easier it gets to do it. That doesn't just apply to having sex, but to murder, rape, doing drugs, stealing, armed robbery, beating, picking a fight, ordering a pointless war, acts of terrorism and just about anything else you are going to have your villain do. Practice makes for callousness, and most of us living in the real world know some of these acts are a one way street; once a character has killed somebody in anger, or in greed, or for the fun of it, we don't believe such characters are redeemable or will ever be safe to leave in society.
If you write such an irredeemable villain, it means if your hero is acting in service to a greater good, they save the future victims of this monster by killing it. Failing to do so is unsatisfying to the reader, because your hero is chickening out, and refusing to make the self-sacrifice necessary to permanently end the threat: The self-sacrifice of shouldering the burden of having ended a life.
I suppose an alternative to killing might be a permanent crippling; leaving them a paraplegic, or some sort of guaranteed permanent incarceration.
The other way out is to just not write such an irredeemable villain. This is possible, I have seen movies where the villain is a just a greedy corporate pig, and doesn't die but ends up behind bars.
But I don't have any deaths in my story, and I'm wondering if this needs to change.
No. You need to check if you have written your villain in such a way that leaving them alive will, inevitably, result in more harm being done to more people.
Otherwise, a good result is "They Get What They Gave". That applies to killing a killer, but could also mean poverty for a robber, abuse for an abuser, slavery for a slaver, etc.
If the villain's actions do not result in death, then death may not be a fair punishment for them. Find a fair punishment that doesn't kill them, a prison, or stranding them on a deserted planet. Disfigurement or blinding (perhaps by accident, which may even have been caused by the villain trying to kill the hero, or unintentionally by the hero in trying to defend themselves from such an attack).
Or have the villain kill themselves: The hero valiantly warns them if they try to escape the Bilbonix will blow, the villain doesn't believe them and tries to escape anyway, the Bilbonix blows up and kills the villain.