Not 100% definitive, but I always thought that he did.
Evidence that he kills her:
Its explicitly stated that he does in the book
The shot of him inspecting his boots on the way out implies that he is checking for blood
movies.stackexchange.com/users/576/kekekela
The cattle gun Chigurh used was with him while he sat frozen like a statue.
Carla wouldn't call the coin toss so he got up slowly and pointed it at her forehead.
Her last words were "now I know why the sheriff told me about this."
He took her car keys, left the cattle gun behind and was on his way to retirement...
sort of...
Chigurh was in a vehicle accident shortly after.
movies.stackexchange.com/users/22071/Ben Johnson 
I googled the question and came upon this discussion and if it's up to speculation then I agree with him checking the boots.
That implies looking for blood and in addition no mentions the car wreck shorty after leaving her house.
If he killed her, he wouldn't want anyone to connect the dots of her murder and the car accident as they were only a few feet apart.
Him walking away versus waiting for medical attention.
Also adds to the fact that he did not want to be caught or connected to that location.
movies.stackexchange.com/users/26352/Jim
The point of that scene is he has no conscious and no qualms killing her.
The coin flip in a way shows he had a hint of remorse about killing her.
He definitely killed her.
movies.stackexchange.com/users/28523/Jeffrey Stein
His coin toss offer was a genuine offer to let her off the hook, just like the guy at the gas station.
The movie has left the outcome to the viewer's imagination so fill in the blank
anyway you like.
I prefer not to have to fill in blanks and am going with what's in the book.
movies.stackexchange.com/users/40282/Juan Lizzo
The "issues" pointing to him killing Carla Jean Moss seem to be:
He's a psychopath/sociopath (sociopaths have a total lack of empathy)
He checks both of his boots when he's "leaving" the house and the that we see inside the house
has no blood up to that point.
His urgency to "leave the scene" points to that.
It's "what he
does"o
She refuses to engage in his coin toss and instead "leaves it up to him" (which, being who he is, was a bad choice) and her intransigence denied her of a 50/50 chance of living, so perhaps the director wanted the double emphasis of her stubbornness and his evil nature to more than suggest that he did kill her.
And, as someone else pointed out, apparently the book makes it clear that...
movies.stackexchange.com/users/48263/Nathaniel Hummel
Chigurh did break into the house and is wanted by the sheriff.
He might want to get away even if he didn't kill her.
The movie is intentionally ambiguous.
Maybe she convinced Chigurh not to kill her.
Maybe he's a changed man.
Odd that he would check for blood on his shoes
(that's why he puts his feet up) if he didn't kill her.
movies.stackexchange.com/users/27916/user27916
I wonder if he did kill her, he didn't kill the two boys who could identify him just down the street from where a body is soon to be discovered?
I think he left her alive and that's why he didn't need to harm the boys either.
The boots being checked on the way out is just a smokescreen.
movies.stackexchange.com/users/47931/John
Towards the end of No Country for Old Men, Anton Chigurh tracks down Carla Jean Moss. They are sitting in a room together and he gives her the option of heads or tails to decide her fate. Is there any definitive evidence that he killed/didn't kill her?
Does Anton Chigurh kill Carla Jean Moss?
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<plot-explanation><no-country-for-old-men>
movies.stackexchange.com/questions/7943
movies.stackexchange.com/users/3203/Travis
In the film it is not 100% definitive but it is explicitly stated that he does in the book. He also checks both his boots for traces of blood while ''leaving'' the house.  There is a lot of evidence proofing that he he killed her
Proof of killing her:
He expressly stated in the book
The shot of him checking out his boots suggests he checks his blood on the way out.  The "issues" pointing to him killing Carla Jean Moss seem to be. He's a psychopath/sociopath. He checks both of his boots when he's "leaving" the house.His urgency to "leave the scene" points to that.