The one thing you can be sure of is that it WASN'T Abner Doubleday.
There are civil war diaries that mention the game ... references in court documents refer to banning "town ball" on certain days back as early as the 1830's....
I don't think the game was actually invented, just like golf wasn't invented. It evolved over time.
I found a reference to "Town Ball" with official rules dated back to 1858 (pre-civil war). Here is the link. You may want to do some research on "rounders" too.
Town Ball Rounders Link If you follow the link, you can see that Rounders was played back in the 1600's ... and it probably has a predessor that has been forgotten.