The man, who gave his name as Arthur Knight, appeared at Edinburgh Sheriff And Justice Of The Peace Court, on Tuesday as part of extradition proceedings against him.
US prosecutors accuse the 34-year-old of the rape of a woman in Utah, as well as attacks on other women in different states.
He is said to have faked his own death and fled to Scotland to escape the charges.
Last week, he was charged with threatening two hospital medics at Queen Elizabeth University Hospital (QEUH) in Glasgow.
On Tuesday, Sheriff Alistair Noble considered a motion to adjourn the extradition proceedings against him until Thursday due to the criminal case in Glasgow.
The suspect was brought into the courtroom in a wheelchair, wearing a maroon pyjama top and trousers. He wore a breathing mask and medics accompanied him with an air bottle.
Sheriff Noble agreed to adjourn the notional hearing to Thursday.
The man, representing himself, then sought to submit his own motion to the court in a bid to restrict press reporting.
He said there was “mass hysteria” around the case, claiming it had “turned into a media circus”.
He said the media had committed contempt of court by “continually referring to me as Nicholas Rossi”.
The man claiming to be Mr Knight told the court: “I would submit that it is incredibly important that this case be adjudicated so as not to include the public humiliation I have faced.”
However, Sheriff Noble said: “I have not heard anything that would justify making any sort of order.”
The sheriff said the matter would be considered by a judge rather than a jury so there was no risk of prejudice to the case.
The man alleged to be Rossi was apprehended in Glasgow earlier this year after being admitted to hospital with Covid-19.
Last month, Edinburgh Sheriff Court was told he was using “every means possible” to delay the extradition proceedings.