In the case of no-fault divorce, if one spouse files for divorce and the other spouse does not want a divorce, then can the other spouse receive protection from the Catholic priest in order to prevent the former spouse from remarriage?
First, I think I should point out that priests have no say in who gets a secular divorce. (Depending on circumstances, it might actually even be the right thing to do.) A tribunal, on the other hand, does have the authority to grant an annulment.
My experience is that the no-fault annulment is possible, but very difficult. It becomes incumbent upon the person who wants the annulment to prove that the marriage is not valid. Sometimes this is easy (one spouse says, "I was piss drunk for our wedding, I slept with her maid of honor (or best man), and I am a schizophrenic who had skipped a month of medication before the ceremony"), other times it becomes almost impossible.
But, that's still "almost". I did have a friend who got an annulment, and her then husband was not too fond of it (though there was clear grounds). He dragged his heals in the process and even managed to delay it for at least a year (the process took three altogether), but eventually she was able to get the Church to hear her out, and the Church agreed with her.
There is a difference between marriage in secular sense and in a religious sense.
The state will allow re-marriage regardless of a persons religion as long as they have followed the state rules regarding divorce and re-marriage. Your priest will have no say in these matters.
Catholics believe that a marriage cannot be dissolved unless there was some issue with form, consent or could not be consumated. Therefore, either spouse cannot marry in the Catholic church without an annulment. The couple could go to another denomination or a Justice of the peace, though.