Is it necessary to renew the marriage contract for a wife who separates from her husband after he pronounces a divorce but it does not take effect?

It is unclear what is meant by "and divorce did not occur." If you meant a false statement that you divorced your wife, while in reality you did not, then divorce does not take place by that, and your wife remains under your marital bond without need for a رجعة (reconciliation) or a new marriage contract.

If you meant that you divorced her but did not officially document the divorce, then the divorce is effective. If it was less than three divorces, you can reconcile with her during her iddah (waiting period) without a new contract, even without her consent. The iddah is three menstrual cycles, or upon delivery of the child, or three months for a young girl or an old woman past childbearing age.

If the iddah expires before reconciliation, then a new marriage contract is required. However, if you have completed three divorces, then she has become irrevocably divorced from you (بينونة كبرى), and she cannot return to you unless she marries another man out of genuine desire, and then he divorces her or dies after consummating the marriage, and her iddah from him ends.