A patient is taken to a hospital's emergency department (the closest one to them) with major trauma and urgently needs surgery for internal bleeding. However, there are no surgeons (for whatever specialty) on call. What will happen here? Is there any way the patient can be stabilized, and then transferred to a facility where surgery can be performed?
In the US this should be a very rare event. Every state (or region) has hospitals designated as level 1 trauma centers. Being certified as a level 1 trauma center means they have all the personnel and facilities necessary to handle any type of trauma at any time. That means surgical teams and all the support staff and facilities such as radiology, blood bank, pharmacy and all the necessary personnel are on site and available 24/7. Surgeons being "on call" isn't good enough.
Along with trauma centers, there is a system of regulations governing which patients qualify as trauma patients and how EMS must handle them. When EMS responds to a 911 call and they determine that the patient meets trauma criteria, they must transport that patient to the nearest trauma center. (This is one of the few times that EMS can overrule a conscious, sober, adult patient's choice of hospitals.)
However, exceptions can and do happen. For example, a patient has suffered massive blood loss. The trauma center is 45 minutes away and the patient is deteriorating rapidly and unlikely to survive the trip, but another hospital is 5 minutes away that has a blood bank and surgical capabilities. In that case the medics would contact their own medical control (usually an ED doctor where they're based) and request a diversion. If approved, they would then contact the new destination hospital to give them a heads up on what's headed their way. You don't take a critical patient somewhere without knowing for sure they can accept them. The idea here is the patient will be stabilized at the smaller hospital and then transferred to the trauma center ASAP.
So the scenario you ask about should almost never happen. There's no benefit to taking a trauma patient somewhere where there are no surgeons since there's little a hospital can do that the medics can't. It's better to fly the patient if possible or simply keep moving. Time is everything in trauma and diverting to a hospital without the necessary resources would be a waste of precious time.
In comments Narusan raises a good question: What if the patient travels to the smaller hospital himself? In that case the same thing would happen. They would stabilize the patient to the best of their abilities and then transfer them to a trauma center as soon as possible.