Do you want a holiday where you can enjoy being alone? Or one which lets you be... alone in a crowd for want of a better word?
If I were looking for the former in the UK, I'd go to the far north of Scotland. Lots of places to escape from everyone, feel "at peace" and all that. Rugged coastline to make you feel small. It's lovely. I'm sure there are places with parallels in the US. Somewhere in the middle of Arizona/New Mexico? Different landscape, similar ends. Oregon? Foresty type wildernesses?
If I were looking for the latter - in the UK, I'd go to London (simply because I meet people I know in Scottish cities. Actually I don't, but I suppose I might).
I love wandering around cities like London - no-one would miss you if you weren't there, but you're never "alone". Again, I KNOW there are places like this in America. I suspect you don't have to look/try too hard to find them. From my sketchy knowledge of a country I've never been to, I'd say New York, Los Angeles, Chicago. I'd want a "smaller" city to get lost in, as opposed to an urban conurbation type place. Seattle, Boston.
I'm probably the worst person to give advice on holidaying alone. I hate it. I want company on a holiday. I had time and money to go on holiday these last summer holidays. I spent AGES trying to persuade, con and otherwise blackmail friends and acquaintances in spending some time with me. In sunny France with decent wine FFS.
In the end, I went with a fab friend who's since moved away, which is awful. We enjoyed nearly a week of flawless driving on the wrong side of the road :S
wine women and song. Or something.
This has turned from advice into a confessional. Apologies.[/code]