My wife thinks I had an emotional affair

Actually, this is about both of you moving on...

I have worked with so many couples where although there was an acknowledgment no sex had taken place with someone else, the thought that 'emotional' communications had been privately shared with a third party was almost as bad and sometimes worse than the worry about sex. That may seem odd but sending and taking messages that you probably wouldn't want a partner to see is likely to cause problems if found out. Often, even the most frantic and sustained denial that 'something happened' is not enough and that’s the situation you face now. 

If you re-read what you wrote to me, you do seem to be admitting that there was some sort of banter between you and the person you met on the hike. Yes, she was with her mother and you were rarely on your own together, however, I think you have to be a bit more honest with yourself. You tell me that the amorous content was mostly from her to you which rather implies you did reciprocate and maybe that felt good. Let's face it, feeling you have a bit of a connection with someone else can be a real ego boost. We can feel more attractive, more entertaining, and more interesting when someone other than a partner makes it clear they may be trying to connect. The problem is of course though that the fallout can be very difficult to overcome. If you think about this from your wife's perspective, finding out that you shared something with someone else that you probably didn't want her to find out about is likely to raise her concerns. 

You haven't included any info about your marriage generally. How have things been between you? Is the current issue so difficult to deal with because there have been other trust issues of whatever description? Are secrets a minefield in your marriage or is there something from way back when that has made this relationship prone to the sort of vulnerability you describe as happening now... 

Of course, it may be none of the above. Your wife may simply be furious with you because essentially, you had something 'private' with another woman, which is for the avoidance of all doubt, exactly what you did. 

But your question to me is what to do now. You pose the question through the lens of your wife needing to move on and presumably forgive and forget. The fact is though, whilst for this marriage to thrive she will need to at least forgive (if not forget), you too need to have a think about what happened. You suggest Relate counselling and that might be a sensible idea but there would be no good attending it if your only purpose is to get your wife to 'move on'.  I suggest this because you and your wife are actually upset about different things. You're concerned that she won't renew her trust in you based on your assertions that nothing much happened with the other woman. She on the other hand is upset because you are minimizing her feelings about the introduction of a third party, no matter how briefly or trivially which has broken her sense of you as a couple. The more you keep telling her there was no sex, the more she will find it difficult to trust you again because put simply, you are not acknowledging how she feels about all this.  

So, you need to start talking differently about what has happened and counselling can be an excellent help in that regard. I would urge you to ask your wife if she might consider this as a first step to seeing what might be possible for both of you to go forward. Making it clear that you want an opportunity to reflect on your actions as opposed to mostly denying them may encourage her to see that you want to build bridges. It really is amazing how once one person shifts a position, their partner sometimes can do the same and that’s the thing that can lead to an open and honest conversation that is often the basis of forgiveness and understanding.