The simplest answer is that while Graves has certainly breached protocol by passing sentence on Newt and Tina without a trial, it's worth noting that Newt basically admitted (in front of multiple witnesses) that he repeatedly and knowingly broke the MACUSA's highest law, something that would typically merit the death penalty anyway.
Rappaport’s Law enforced strict segregation between the No-Maj and wizarding communities. Wizards were no longer allowed to befriend or marry No-Majs. Penalties for fraternising with No-Majs were harsh. Communication with No-Majs was limited to that necessary to perform daily activities.
and
A significant difference between the wizarding governments of the United States and the UK of this time was the penalty for serious crime. Whereas British witches and wizards were sent to Azkaban, the worst criminals in America were executed
When the 'executioners' look a little askance at the speed of his decision, he simply shrugs it off. He acknowledges that he's breaking with protocol and just says that he'll get it sorted. They evidently trust him to do so.
GRAVES: (to the EXECUTIONERS) Just do it immediately. I will inform President Picquery myself.
Since Graves is in full charge of wizard security, second only to the President, his decision is certainly unusual but it's not surprising that his placemen did exactly what they were told. It's also possible that he has some kind of emergency powers that he can use in cases like this where there's an ongoing risk.