I think that it would actually be an issue to decide by the host country.
Since it would be the host country's responsibility to detain and extradite Assange, it would be up to it to decide the standing of the USA requests vs Assange's rights.
If we are talking about Rule of Law countries1, then it is up to its judiciary to decide if an extradition would be a violation of the subject rights or not. Which means that usually some charges should need to be filled in the host country legal system, and the judge should have to decide on its merits (they are not politically motivated, the accused may get a fair trial, that there is enough evidence for a trial, etc.).
So, if the USA asked for an extradition it would not be illegal for it to be without charges, but most2 Rule of Law countries could not proceed with it without violating Assange's rights. But that would be a violation made by the host country, not by the USA3.
And of course, if the petition were to succeed, once Assange arrived to the USA charges would have to be filed at that moment; otherwise he could ask for habeas corpus and challenge any detention4.
TL/DR The USA may ask anything they want, it is up to the host country to decide which rights Assange have and to uphold the rule of their law.
Countries without rule of law are easier, as they procedure is basically what the rulers say.
2I am tempted to say "all", but I am really not sure.
3Apart from the local legal system, there are international laws that would enter into effect if Assange could show that he was to be tortured, unjustly prosecuted, etc. in the USA. But even those issues would be addressed by the host country legal system.
4Assuming he is not put in the care of a Guantanamo Bay style "legal shortcut".