What is the ruling on breaking a promise made to Allah Almighty, knowing that the promise was to stop watching lust-inducing websites?

The broken promise to abandon the sin should have been fulfilled. One must repent to Allah for breaking the promise and committing the sin. This promise is not considered an oath, and therefore no expiation is required. As long as the repentance is sincere, Allah is Forgiving and Merciful, and no sin is too great for Him to forgive. "One who repents from sin is like one who has no sin."