As others have said, it sounds like a crush in which he had idealized her and is craving her attention. The more you nag him about it the more he will be silent with you. He’s juggling two women in his head - the vivacious, popular, fantasy woman across the road and his comfy wife of many years who is criticizing him. This makes you the enemy because you are calling him out and embarrassing him over something that he may feel embarrassed about anyway. Any slight reciprocity from your neighbour will give him hope, even one innocuous reply to his ten messages.
I disagree with others who say talk to her because if she is enjoying the attention then it might increase the frisson for her if she knows it’s upsetting you. If she feels uncomfortable she can let you know. Depending on what kind of woman she is, she and her friends might be having a good laugh about this. Silly old s*d who lives across the road with his boring wife so he's got the hots for her. You don't want to be rising to that kind of thinking.
I was widowed young and have lead a single and very independent life for the last twenty years. I have been on the receiving end of a fair number of crushes from men of all ages, some of them married, some of them considerably older than I am. There’s something about a single, independent woman that acts like catnip. Motivations are different. Some think single women are fair game, some simply admire that independence, others want to be helpers and protectors. It sounds like your husband may be in the latter category.
My advice is to do nothing. Don't nag him, don't speak to her about it. Just ride it out until he loses interest, which he will, in time.