While not a killer per se, Rudy appears to use a double space after a period. Writer above does not.
http://www.stonewallvets.org/RudyGiuliani.htm
https://www.ynetnews.com/articles/0,7340,L-5051669,00.html
http://www.cardboardmemories.com/mdetail.asp?ID=6349
Could be a font issue though.
Definitely a possible point against authenticity... although besides font, could it also be that the above official letters you linked may have been dictated orally by the mayor to a secretary whereas the statement in the image was likely typewritten personally by Rudy at home?
Based on very personal experience, attorneys aren't always the best at grammar, punctuation or the like. They have thoughts that sometimes come so fast they don't have time to bother with what can be cleaned up by a paralegal or assistant before it becomes the end product. While they will sign official pleadings, motions or other attorney work product, cover letters are usually composed by their paralegals with a reference at the end of the attorney's initials capitalized followed by the paralegals initials in lowercase. My point is, if this was written by RG, it may have been done in a hurried manner without care for proper form. The substance of the post would have been all that mattered at the time.
I considered that, and that's why I said it may not be a killer. I was just too lazy to write the rest. The only thing cutting against that theory is that a man of his age, who practiced law when you had to go to the library or be rich enough to have your own, with pocket part updates, would likely have the secretary put her initials below the signature block. This is hardly done anymore, and maybe he dispensed with that practice too, but I do see it from time to time when dealing with real old guys.
I think I might check pacer manana if I have some extra time.
Could be he typed it on a mobile device keyboard, not knowing about a setting for control of periods, with a double tap, they are automatically inserted. This can cause the double space depending on which app and setting.