It's not inappropriate at all. As one who loves conversation over meals, eating alone takes away from the ambiance of dining out and truly enjoying the prepared meal. I've met women that have no problems dining alone. I view it as someone who's there to enjoy her situation, even if it means doing so alone.
Is a woman sitting alone in a bar/restaurant somehow inappropriate?
Not in America, not anymore.
Maybe in some backwater town deep in the bible belt, but just about everywhere else you should feel comfortable doing so.
It is your right to do so.
If men do come up to you, take it as a compliment. Should you desire your privacy, a gentleman will respect your request for the same, whereas cads will probably make a snide remark as they walk away.
Having said that, when I am sitting in a strange location by myself, I often (not always) wish that a woman would come up and simply say "hello, may I join you?".