Qualifier: I work ER/Trauma, and have worked too many Gun Shot Wounds (GSWs) to count.
Articles for you, OP: Here's one in the Atlantic That states the same thing as the WaPo article that /u/TiberiCorneli quotes. An article by the Baltimore Sun states that "survival" rates for a single GSW to the head is ~ 5%.
Some notes - The "brainstem" is talked about here in my response, brainstem is the lower part of the brain found near the base of the skull that controls basic functions, most importantly in the scenario, those of cardiopulmonary maintenance (heart and lungs).
Let us move to a medical source: a public page from UCLA Neurosurgery has some good data in it that tells what things may have looked like had Lincoln been rapidly evacuated to a Level I Trauma center and aggressively resuscitated.
Treatment
Virtually all cranial gunshot victims are aggressively resuscitated upon initial arrival at the hospital. The decision to proceed with surgery to manage the wound is based on three factors:
The level of consciousness as determined by the Glasgow Coma Scale (GCS)
The degree of brainstem neurological function
The findings on the CT scan
Death is virtually certain among comatose patients with minimal evidence of brainstem function and no evidence of an intracranial hematoma.
Lincoln likely would have been considered comatose: "A 23-year-old doctor named Charles Leale was in the audience and hastened to the presidential box immediately upon hearing the shot and Mary Lincoln’s scream. He found the president slumped in his chair, paralyzed and struggling to breathe.".^[2] Brain stem function is difficult to assess, but we do know that Lincoln was shot at 10:15 pm (2215), and was declared dead at 7:22 am (0722). ^[2] This gives us hope that he was "alive" in the sense that his brainstem was still able to keep his cardiopulmonary system functioning. But was he interacting? He was declared ~ 9 hours after his trauma; Manytimes GSW will bleed out rather rapidly.
It is feasible that the single, low velocity GSW (although large caliber {.44}, it was fired from a deringer with a 2 1/2 inch barrel, using black powder) may not have killed Lincoln, and that he could have survived his wound. Please note that as you will read below, there was no exit wound, the bullet stayed in the brain. This is an indicator of the lower power of the round; a modern 9mm, .40caliber or .45ACP can quite often leave an exit wound in a cranial GSW.
Also, something to consider for Lincoln post-GSW is quality of life. Quite often, when a GSW injures higher functioning sections of the brain and does not cause a massive hole in the tissue (which happens with higher velocity rounds like rifle rounds), or a bleed in the brainstem, you will find that the patient has long lasting sequelae of a TBI (traumatic Brain Injury) - behavioral problems, cognitive deficits, logic faults. What would this have meant for a recovering Lincoln? Years of rehabilitation, and likely no return to being the PotUS. A good analogy is Congresswoman Gabrielle Giffords who sustained a single non-transecting GSW to the head. Please note that "she was still conscious and "following commands" at the time"^[1] {of her arrival to the Trauma Center} which puts her at odds with Lincoln, who was abtunded (non responsive and laid out) almost immediately. Also, look at James Brady, who retired after recovery from his GSW to the head. He was never able to work again in a political capacity.
UCLA Neuro talks about outcomes:
Outcome
Recovery after brain injury varies widely. The predictors of poor brain function outcome or death after a gunshot wound to the head include the initial GCS score, older age, presence of low blood pressure or inadequate blood oxygen shortly after injury, and dilated non-reactive pupils.
The bullet trajectory through the brain carries major significance. Bullets that cut through the brainstem, multiple lobes of the brain, or the chambers where spinal fluid is located are particularly lethal.
Transecting GSWs (getting shot in the side of the head vs. the front or back) have a much higher rate of death then anterior or posterior gunshots. Lincoln was shot from directly behind: "The ball entered through the occipital bone about one inch to the left of the median line and just above the left lateral sinus, which it opened. It then penetrated the dura matter, passed through the left posterior lobe of the cerebrum, entered the left lateral ventricle and lodged in the white matter of the cerebrum just above the anterior portion of the left corpus striatum" ^[3] FYI, that "left lateral ventricle" is one of those "chambers where spinal fluid is located"; this is lethal because of the increased bleeding into an open space. The brain is very vascular.
Many initial survivors develop uncontrollable pressure on the brain and subsequently die.
This is true. In ICU, the patient will quite often suffer a rapidly rising ICP (IntraCranial Pressure) due to the body responding to the trauma, and will die despite the latest interventions in pressure management. That high pressure also leads to diffuse axonal compression and anoxia, meaning more brain damage.
TL;DR: He might have survived, it sounds like a low chance of it; and he likely never would have been president again.
SOURCES
^[1] Gupta, Sanjay (January 10, 2011). "Gupta: What helped Giffords survive brain shot."
^[2] ABRAHAM LINCOLN’S ASSASSINATION
^[3] The autopsy of President Abraham Lincoln from [http://www.nlm.nih.gov/visibleproofs/galleries/cases/lincoln.html]
^[4] "Prognostic factors and treatment of penetrating gunshot wounds to the head". Roberto S Martins, M.D., , M.G Siqueira, M.D., M.T.S Santos, M.D., N Zanon-Collange, M.D., O.J.S Moraes, M.D.* Surgical Neurology Volume 60, Issue 2, August 2003, Pages 98–104
^[5] "Analysis of 76 civilian craniocerebral gunshot wounds" W. Craig Clark, M.D., Ph.D., Michael S. Muhlbauer, M.D., Clarence B. Watridge, M.D., and Morris W. Ray, M.D. Journal of Neurosurgery July 1986 / Vol. 65 / No. 1 / Pages 9-14
Post removed as it's not a history question. For a discussion of modern medical treatment, try x-posting to /r/AskScience.