The nuclear family used to be one of the strongest aspects of America. It united our citizens and allowed us to rely on family rather than the government. Children raised in single parent households are far more likely to end up in prison, addicted to drugs, face more economic hardships in their lives, are far less likely to attend college. No fault divorce perpetuates economic hardship for American citizens.
Given that our culture no longer sees marriage as a base requirement for intimate relationships or even children, and given that the economic barriers to women that made marriage one of the few options to avoid poverty have been removed, my concern would be that getting rid of no-fault divorce would actually lead to marriage becoming an outdated practice that few participate in in an effort to maintain control over the nature of relationships.