In the book when speaking of his wife they say that Goldsmith's "Rome" would have been a staple. I can find no book from the 1700's or around that time by Rome. Can anyone shed some light on this book?
I don't have Chernow's book, but searching the time, author and subject matter on Google Books, I believe what you're looking for is The Roman History available here.
The book went trough a number of subsequent reprintings and abridgements into the 19th century, and a few references can be found made to it, and the author's other works, in the US National Archives' Founders Papers, such as this list of books for Congress from 1783.
A later version, mentioned in an 1821 letter from James Madison, was published in Philadelphia in 1795, so it was clearly known in the early United States.