This is an interesting question. Firstly, Stephen is not the only other claimant to the throne in 1135, though he is the principle one. The Norman nobility had considered naming his elder brother, William I's eldest (mentally able, legitimate) grandson, Theobald, Count of Champagne, as king, and if Stephen does not press his own claim then perhaps Theobald can arrive in England before Matilda does in order to try and take the throne for himself. Given that Theobald accepted the crown from the Norman nobility it seems that the desire is there, even if he does lack some of the conniving and drive of his younger brother. Theobald has the benefit of having three sons in 1135, compared to Stephen's one.
So it is possible that England would not avoid a civil war. It is difficult to say how the Anarchy might go with Theobald fighting Matilda instead of Stephen (of course, we can expect Stephen still to have some involvement supporting his brother, much as Theobald did). However, one of the key factors in ending the civil war was the death of Eustace of Boulogne, leaving Stephen without a son old enough to strongly claim the throne. As mentioned previously, Theobald is in a stronger position and is less likely to be left without an heir of sufficient age, and so can continue fighting. Additionally, given that he had a great deal of support among the Norman nobility, this could deny Matilda her main base of power (excluding Anjou), and thus prevent her drawing the civil war out long enough to force a favourable peace (as she did OTL).
If Theobald keeps the throne out of the hands of Matilda and her descendants (as I am suggesting is possible) it sets a dangerous precedent for the succession. Are women to be able to ascend to the throne in the absence of male heirs (agnatic-cognatic primogeniture)? Does succession pass through the male line only (agnatic primogeniture)? Does the succession exclude females from inheriting but is able to pass to a male through the female line (uterine primogeniture - the method by which Theobald pressed his claim)? And for how long will Theobald and his heirs be able to prevent Matilda and her descendants from attempting to press their claims again, assuming he does not remove them from Anjou? It would seem that England would be destined to return to civil war before too long - perhaps soon, when the Angevins attempt to press their claims, or later if there are question marks over the succession.
If Matilda did take the throne unopposed, I think her rein may have been somewhat disastrous. Matilda showed little talent for governing when she was presented with the opportunity, and also had an unfortunate habit for annoying the barons.
With dissatisfaction at her methods, and a general distaste for having a female monarch, the barons might have tried to get some sort of Magna Carta-type document forced upon Matilda in order to curtail her power. This is not a far-fetched idea, as the barons in 1215 took their inspiration from 1100's Charter of Liberties, which would have been far fresher in the minds of the barons, given that it would have only fallen by the wayside in 1135 (her character suggests that Matilda would not have allowed to herself to be bound in the way that the Charter of Liberties had her father, which is the same response that Stephen had).
If the barons did not attempt a legislative solution, then there are always two counts in France with claims to the throne who might prove better monarchs than Matilda...