Republicans want to make it illegal or at least much harder to obtain a no-fault divorce because they see it as harmful to society in that it undermines the sanctity of marriage and the family. I agree that divorce in general does this and that Jesus' teaching on marriage and all His teachings are still relevant today.
HOWEVER, the error the republicans are making is that they view the U.S. as a historically Christian nation as if it were in a convenant with God like Israel was in the OT. They presume that they then must influence the government to enact laws based on their religious beliefs. This then forces people who do not have faith to obey a God they do not know. The U.S. is NOT a theocracy. Forcing people to obey moral laws they do not believe obtains a sort of conformity in society, but does not have the power to change a person's heart. It's a legalistic attempt to conform a person's behavior. The person may have outward obedience, but there will be resentment and no inward transformation on their part.
Paul writes in this way: "Bondservants, obey your earthly masters with fear and trembling, with a sincere heart, as you would Christ, not by the way of eye-service, as people-pleasers, but as bondservants of Christ, doing the will of God from the heart, rendering service with a good will as to the Lord and not to man..." [Eph 6:5–7]
It is interesting that slavery is mentioned in this post as an ill to be dealt with in society. Paul speaks here of slaves, but ALSO their believing masters in the next couple verses: "Masters, do the same to them, and stop your threatening, knowing that he who is both their Master and yours is in heaven, and that there is no partiality with him." [Eph 6:9]
Jesus DID teach and preach against the root cause of slavery, pride: "And Jesus called them to him and said to them, "You know that those who are considered rulers of the Gentiles lord it over them, and their great ones exercise authority over them. But it shall not be so among you. But whoever would be great among you must be your servant, and whoever would be first among you must be slave of all. For even the Son of Man came not to be served but to serve, and to give his life as a ransom for many."" [Mar 10:42–45]
I'd also like to argue that Jesus' preached and taught against a greater evil than physical slavery. He taught that we are all in spiritual slavery to sin, and that only He can make us free: "Jesus answered them, "Truly, truly, I say to you, everyone who practices sin is a slave to sin. The slave does not remain in the house forever; the son remains forever. So if the Son sets you free, you will be free indeed." [Jhn 8:34–36] True faith in Christ transforms a person on the inside. All other religions try to conform a person from the outside.