Much like there wasn't one "Caesar", since it's a title (still kicking in Proto-West Germanic derivatives branching off "kaisar"), but many "normal" people (not history nerds) would still use only the title to refer to Gaius Julius Caesar.
Also fun fact -- if you're a history nerd -- "Julius Caesar" is almost equally nonsensical to just "Caesar" since "Julius" is not his name, but refers to his family ("gens"). The first Caesar from that family, that we know of at least, was Sextus Julius Caesar in around 200 BC, 300 years before Gaius Julius Caesar was born.
Also fun fact -- if you're a history nerd -- "Julius Caesar" is almost equally nonsensical to just "Caesar" since "Julius" is not his name, but refers to his family ("gens"). The first Caesar from that family, that we know of at least, was Sextus Julius Caesar in around 200 BC, 300 years before Gaius Julius Caesar was born.