Full disclosure: I'm 3 things which may not endear me to you folk - a guy, a dad and a guitarist. Also, this is going to be an epic post, because this sort of situation bothers me and causes strife where there really doesn't need to be any.
However...I thought I'd try to help here, because I may even have something useful to say ;)
It's often not quite as simple as "using headphones" - as has been pointed out, he's following YouTube videos. That means he has to be use something which allows him to hear both the guitar (the amplified sound, or something like it) and the video itself. While this is possible (and I'll tell you how in a sec), plugging headphones into his amp is not going to to it. In fact, most amps don't have a headphone output (plugging them into the speaker output would fry the headphones and release an awful lot of smoke) and when they do the sound is often so bad it's like somebody shoving ice picks into your brain via your ears. Trying to practice like that is impossible (I know, I've tried).
Thing is, he probably isn't even aware of a way to have all the audio run through the computer so he can hear both his playing and the video through his headphones.
There is, however, a way to do it but it's going to cost a bit of money. Firstly, he needs to get a decent low-cost audio interface (M-Audio Fast Track should do it - these are around £50 if you shop about a bit - and they're easy to set up). There are much better, and much more expensive, interfaces about but if he's just using it for practice then that shouldn't be necessary.
Secondly, he'll need a bit of software on the computer which will take the raw sound of the guitar and make it sound like it's running through an amp - Amplitube Custom Shop is free to download and install, and comes with a few decent amp models and effects. It may even sound better than his current amp. He could also buy new models for it in the Custom Shop part, so he can sound more like his heroes :)
An extra bonus of this is that he could use this hardware and software to record himself playing. Listening back to that could well help him improve his playing - that's not me saying that he's crap, just that there's always a way to improve no matter how good you are.
So that's the helpful bit out of the way...now for the opinion (I'm bracing myself already). It sounds to me like you're both being unreasonable and insensitive. He's ignoring everybody else in the house by retreating into his world while he's playing. While every musician does this, it's pretty inconsiderate and - generally - shouldn't be done, because even from a purely selfish perspective the inevitable friction is never going to be conducive to good practice.
However, the OP is also being pretty insensitive. Several things are fairly obvious:
1 - He loves playing guitar.
2 - He may not be of a high standard (playing Brown Eyed Girl repeatedly to get it right suggests limited ability).
3 - He's very aware that he's not playing at a high standard, but lacks the guidance to get better.
4 - He doesn't know how to improve, and is getting frustrated.
As his wife, you have a reasonable expectation that he be more considerate, but he should also have a reasonable expectation that you would also have spotted points 1-4 above since you're supposed to know him better than anybody else.
My suggestion - tell him you've researched it with musicians, and buy him the audio interface I mentioned above. It's pretty cheap (as these things go), and it will be a step on the way to him getting what he wants - being a better player - as well as you getting what you want.
Approached the right way, the compromise is on both sides: yours is purely financial, and his is in changing the way he does things (to what will - at first - appear to be a worse setup) to make you happier.