Could Peeves have gotten thrown out of Hogwarts?

Question

Sorcerer's Stone:

About twenty ghosts had just streamed through the back wall. Pearly-white and slightly transparent, they glided across the room talking to one another and hardly glancing at the first years. They seemed to be arguing. What looked like a fat little monk was saying: "Forgive and forget, I say, we ought to give him a second chance --" "My dear Friar, haven't we given Peeves all the chances he deserves? He gives us all a bad name and you know, he's not really even a ghost...

What would the ghosts have done to Peeves if they decided to crack down? The quote implies they had some way of punishing him - presumably expulsion - particularly when (presumably the Bloody Barron) that says

haven't we given Peeves all the chances he deserves?

Could the ghosts have thrown out Peeves from Hogwarts, and if yes, how do they have that power?

Answer 1

Almost certainly not.

JK Rowling has described Peeves as a kind of permanent feature within Hogwarts that cannot be removed.

ES: Why does Dumbledore allow Peeves to stay in the castle.
JKR: Can't get him out.
ES: He's Dumbledore, he can do anything!
JKR: No, no no no no. Peeves is like dry rot. You can try and eradicate it. It comes with the building. You’re stuck. If you've got Peeves you're stuck.
ES: But Peeves answers to Dumbledore -
JKR: Allegedly.
MA: Allegedly?
JKR: Yeah. I see Peeves as like a severe plumbing problem in a very old building, and Dumbledore is slightly better with the spanner than most people, so he can maybe make it function better for a few weeks. Then it’s going to start leaking again.
_{(Leaky Cauldron interview).}

Peeves is by his very nature difficult to eject.

...it was likewise to be expected that such a poltergeist would be noisier, more destructive and harder to expel than those that occasionally frequent Muggle houses.
_{(Pottermore, "Peeves").}

After all, the last attempt to get rid of him didn't exactly go well.

Several concerted efforts to remove Peeves from the castle have resulted in failure. The last and most disastrous was made in 1876 by caretaker Rancorous Carpe, who devised an elaborate trap, baited with an assortment of weapons he believed would be irresistible to Peeves, and a vast enchanted bell jar, reinforced by various Containment Charms, which he intended to drop over the poltergeist once he was in place. Not only did Peeves break easily through the giant bell jar, showering an entire corridor with broken glass, he also escaped the trap armed with several cutlasses, crossbows, a blunderbuss and a miniature cannon. The castle was evacuated while Peeves amused himself by firing randomly out of the windows and threatening all and sundry with death.

A three-day standoff was ended when the Headmistress of the day, Eupraxia Mole, agreed to sign a contract allowing Peeves additional privileges, such as a once-weekly swim in the boys' toilets on the ground floor, first refusal on stale bread from the kitchen for throwing purposes, and a new hat – to be custom-made by Madame Bonhabille of Paris. Rancorous Carpe took early retirement for health reasons, and no subsequent attempt has ever been made to rid the castle of its most ill-disciplined inhabitant.
_{(Pottermore, "Peeves").}

Most of the attempts seem to have been made by humans so there's no confirmation on whether or not the ghosts would've been able to make good on their threats. Peeves only really answered to Dumbledore and the Bloody Baron, and even then in moderation.

No attempt was made to expel Peeves by the ghosts so we don't know if it would've worked. Maybe Peeves would've left if the Baron got pushy. Maybe he wouldn't. But it seems that they couldn't force him to leave if he didn't want to.

Answer 2

Filch, Umbridge, and Fudge apparently thought Peeves could be expelled, as we find towards the end of Order of the Phoenix:

But when Educational Decree Twenty-nine comes in, Potter, I’ll be allowed to do them things... And she’s asked the Minister to sign an order for the expulsion of Peeves... Oh, things are going to be very different around here with her in charge..."

Filch had also expressed the possibility of expelling Peeves back in Goblet of Fire, when he found Harry's egg:

"Hiding, are you?" he said softly. "I'm coming to get you, Peeves.... You've gone and stolen a Triwizard clue, Peeves.... Dumbledore'll have you out of here for this, you filthy, pilfering poltergeist...."

"The thing is, Professor," said Filch plaintively, "the headmaster will have to listen to me this time. Peeves has been stealing from a student, it might be my chance to get him thrown out of the castle once and for all —"

Fleur Delacour also assumed that a poltergeist could be expelled, judging by her statement to Roger Davies at the Yule Ball in Goblet of Fire:

We 'ave none of zis ugly armor in ze 'alls, and eef a poltergeist ever entaired into Beauxbatons, 'e would be expelled like zat." She slapped her hand onto the table impatiently.

However, the mere fact that various characters believed it to be possible doesn't necessarily mean that it actually is possible.

On the other hand, it would make sense that Peeves could be expelled. We know from Percy's comments when they go to Gryffindor Tower for the first time in Philosopher's Stone that Peeves more-or-less listens to the Bloody Baron:

"You want to watch out for Peeves," said Percy, as they set off again. "The Bloody Baron's the only one who can control him, he won't even listen to us prefects. Here we are."

If there was nothing the Baron could do to Peeves, Peeved would have no reason to be afraid of him.

Additionally, there was another discussion between the ghosts about Peeves prior to the opening feast in Goblet of Fire:

"Peeves, of course," said Nearly Headless Nick, shaking his head, which wobbled dangerously. He pulled his ruff a little higher up on his neck. "The usual argument, you know. He wanted to attend the feast — well, it's quite out of the question, you know what he's like, utterly uncivilized, can't see a plate of food without throwing it. We held a ghost's council — the Fat Friar was all for giving him the chance — but most wisely, in my opinion, the Bloody Baron put his foot down."

This implies that there is some method that is capable of keeping Peeves out of the Great Hall, for otherwise Peeves could have just entered despite the protestations of the ghosts. If there is a way to keep him out of the Great Hall then there should also be a way to keep him out of Hogwarts entirely. If there was no way to physically stop Peeves from entering the Great Hall then there would be little reason for him to not enter out of fear of the Bloody Baron, because there would then seem to be nothing the Baron could actually do to him.

Furthermore, we know that spells do work on Peeves. During the first Defense Against the Dark Arts Class in Prisoner of Azkaban Lupin uses a spell on Peeves when he refuses to obey him:

He raised the wand to shoulder height, said, "Waddiwasi!" and pointed it at Peeves. With the force of a bullet, the wad of chewing gum shot out of the keyhole and straight down Peeves's left nostril; he whirled upright and zoomed away, cursing.

If Peeves is susceptible to such magic it would stand to reason that there is some magic that would be capable of removing him from the castle. We certainly see Peeves physically interacting with things throughout the books, so it should be possible to physically move him.