The England forward posted a message on the social media website on Friday that included the line "Man was made for woman to procreate that was the goal no?".
Saracens, who held "lengthy discussions" on Sunday with their number eight, branded Vunipola's actions "a serious error of judgement" on Monday evening.
A club statement read: "At Saracens, we are one family, open to all with the firm view that everyone should be treated equally with respect and humility.
"We recognise the complexity of different belief systems and understand Billy's intention was to express the word of God rather than cause offence.
"However, he made a serious error of judgement in publicly sharing his opinion, which is inconsistent with the values of the club and contravenes his contractual obligations.
"The player has been formally warned about his future conduct."
Vunipola insisted that his "intention was never to cause suffering" but he stopped short of offering an apology.
In a statement on Saracens' website, he said: "I can see that my recent post has hurt people.
"My intention was never to cause suffering. My intention was to express my belief in the word of God.
"These beliefs are a source of great strength, comfort and guidance in my life. This is deeply personal and does not represent the views of my team.
"Anyone who knows me, knows I live with kindness and love towards ALL people.
"I have faith. My Christian faith has sustained me through the ups and many downs of my life. God has given and continues to give me many reasons in life to be grateful."
Vunipola has also been summoned to talks with the Rugby Football Union but the governing body is refusing to provide any details for its meeting, insisting it remains a private matter.
Twickenham has also been wary at this stage of saying whether disciplinary action will be taken, but if it is decided to pursue the matter further, it would most likely be under the broad charge of 'bringing the game into disrepute'.
Condemnation of his views has also come from Premiership Rugby and the Rugby Players' Association, while on Saturday Vunipola was booed during his appearance as a second-half replacement at Bristol.
Rugby Australia confirmed on Monday morning that it is pressing ahead with plans to terminate Folau's contract after its integrity unit ruled he had committed a "high-level breach of the professional players' code of conduct".
A legal battle with RA could ensue as Folau is refusing to retract divisive remarks which have been liked by a number of international players, including Vunipola.
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