Utilitarianism can exist alongside Christianity because both accept the equality of the individual. The difference is that the outcome in Utilitarianism is a mass decision as a product of individual and equal voting.
Utilitarianism accepts the fact that in order to achieve the greatest good, there will be victims as a result. This would go against the idea of ‘agape’ and loving your neighbour. Christians believe that it is their responsibility to look after and help the weak, not to simply accept and victimise them. The Golden Rule is ignored by Utilitarians; to love one’s neighbour as oneself is not to treat them as one of many, although Mill’s harm principle does go some way towards this fundamental Christian teaching.
There is a Christian/ Utilitarian contradiction when it comes to consequences. This is because Christians believe that only God can judge and decide the consequences- not humans.
However, the two can co-exist if it is recognised that Christians are good in life in the hope that the consequence is going to heaven. Surely this is the same as Utilitarianism? (teleological?)
Bentham was criticised for being a hedonist i.e. all that mattered was resulting pleasures. Mill expanded on this to say that the more important (higher) pleasures were intellectual as opposed to physical. However, a Christian would argue that happiness on earth is irrelevant. What is important is spiritual happiness and happiness in the afterlife.
Rules also cause contradiction between Utilitarianism and Christianity, particularly between Rule Utilitarianism and Christianity. A rule Utilitarian would say that the greatest good for the greatest number is the most important rule; Bible teachings, such as the 10 commandments are helpful but can be broken to create the greatest good and this goes against a Christian viewpoint. It would be easier for Act Utilitarianism and Christianity to co-exist. E.g. Situation Ethics is a Christian