I recently read a great question on workplace.se and it reminded me of a similar situation I've come across a couple of times, where I did not know how to interrupt the speaker without embarrassing him even more. It often happens with non-technical coworkers that misunderstood something technical, but can also happen in a situation between friends.
I think it would be easiest to take the example of the workplace.se question:
John starts his argument with how he found out about the 'new mom's room' that was recently created. He is disappointed there is no new dad's room as newborn baby life is just as hard for dads as it is for moms. He goes on about how unfair it is new moms 'get to nap' and that change needs to be made. The company needs to be more mindful of equality, so a new dad's room needs to be made too, so dads can unwind or have a power nap too.
Now John here has misunderstood the meaning of this 'new mom's room'. The room was intended for breast feeding women to be able to have privacy while pumping. I understood this quite early on in his story and could have stopped him from embarrassing himself any further. With this topic he could potentially offend people as well.
How do you stop someone in a public or group situation like this, without putting his mistake in the spotlight even more?
Or would putting the mistake in the spotlight give him a chance to set his argument straight?