In his time, he was referred to as Gaius Caesar or simply as Caesar by himself and his contemporaries. Calling him “Julius Caesar” would have been redundant since every Caesar was a Julius.
The Romans had a lot of traditions around names. Gaius Julius Caesar inherited his name from his father, grandfather, and great-grandfather.
Caesar’s great-nephew Gaius Octavius Thurinus also took the name Gaius Julius Caesar when he inherited the assassinated dictator’s estate. To prevent confusion, he’s often called Octavianus, which is more or less the ancient Roman equivalent to “the artist formerly known as Prince”. The emperor Caligula, Octavian’s great-grandson, was also named Gaius Caesar; we know him by a nickname he received as a small child.