Dr. Armstrong might just be the second most upstanding guest at the island. A respected doctor, he can’t get enough of the fact that he’s led a fine professional life: “He’d been lucky! Lucky and skillful, of course. He was a good man at his job” (1.68).
Well, except—you knew there was an “except” coming, didn’t you?—for the “incident” that happened fifteen years earlier, where he killed a woman on the operating table. Yeah, except for that. Oops.
Still, he clings to the idea of himself as a respectable member of society. When the guests at Soldier Island are trying to figure out who the killer is, he rolls his eyes at accusations, saying that he’s a respected member of society and therefore could not be the killer. Obvs. Because no one respectable has ever done anything bad, ever.
Thing is, this inability to see beyond “respectability” that dooms Dr. Armstrong in the end. He trusts Justice Wargrave because he is a judge and therefore above reproach. When Justice Wargrave convinces him to go along with the faked death plan, Dr. Armstrong readily helps, even though it will only lead to his death. We’ll let Wargrave explain:
According to my plan I should shortly need an ally. I selected Dr. Armstrong for that part. He was a gullible sort of man, he knew me by sight and reputation and it was inconceivable to him that a man of my standing should actually be a murderer! (E.168)
In other words, Armstrong’s respect for those in power makes him exactly gullible enough to be a prime victim of Wargrave’s evil genius. He becomes nothing more than a pawn in the whole plan: when the other characters become convinced that he must be the killer since he disappears in the middle of the night.
When his body shows up, it becomes clear: Armstrong trusted the wrong person for the last time.