ReallyConfused, sorry to say this but he is quite clearly having an affair.
I understand the reasoning behind your question "why doesn't he just come clean and admit it?", but I can offer the other side, if you like.
I had a fling with an old friend 2 yrs ago. It only lasted about 2 weeks, and during that time I only actually saw him twice, as he lived so far away. But the "passion" was maintained through constant mobile calls and texting. My dh found out when he read my mobile. I had not really given any thought to why I was doing what I was doing at the time (other than I felt very unhappy) and I certainly hadn't made up my mind that my infidelity meant I wanted out of my marriage. As long as dh didn't know about it, the choice was mine to make, I had all the power. Once he did know about it, he could quite reasonable leave me whether i wanted him to or not.
So when he confronted me with this bill, and asked me what was going on, I dug myself in deeper and deeper by trying to say that yes, we were flirting a bit, but nothing had happened...
of course he dragged it out of me in the end. I then felt so guilty for lying on top of what I had done, I bent over backwards trying to persuade him that we cld make a go of it.
We stayed together but 2 yrs on, all the reasons why I was unfaithful are still there, and I think we'll split up anyway (although I'm not being unfaithful now, and wouldn't again anyway).
My point is, infidelity doesn't have to be the end of a marriage, but in order for things to improve, both parties need to take it seriously and want to work through it. If he cannot bring himself to tell you the truthm even when it is staring you both in the face, then IMO he must either be very stupid himself, or have absolutely no regard for you and your feelings.
If you want to work it out, I think you have to just be very cold and firm and explain calmly that you know what is going on, and he can either 'fess up like a responsible adult, or move out. Don't let him treat you like this, you deserve better.