That's not how the incentives align. If Visa is responsible for eating fees from fraudulent charges, they're going to start removing vendors with higher than usual fraud charges from the network. Businesses will still have an incentive not to allow fraud, because they won't be able to process credit cards otherwise. This already exists for chargebacks, and businesses hate them because too many chargebacks increases the percentage the processor takes.
Counterfeit notes are different. There is no information asymmetry. What makes a dollar bill a dollar bill is widely published, so validating the authenticity of one is rather easy. There are tons of companies that make machines that will tell you if your dollar is real or not.
Credit card charges are different in that there is a huge information asymmetry. Your credit card network knows who you are. They have your address. They have a copy of your usual spending habits. They know who lives in your house. They have a ton of information they can look at to see whether a given charge makes sense. They can add additional security measures.
You know what your merchant knows about you? That you want to buy X, and you're paying with Y credit card. They don't have any meaningful information to make a judgement about whether this purchase is legitimate. I struggle to think of how a merchant legitimately has the option to stop fraud. You can pay one of those anti-fraud sites, but at that point you're really just paying so you can get access to the same information that the credit card processor already has.
Some of the CC processors even already have an anti-fraud service. It sure seems like they're the best poised to offer it, they just want to be able to charge you more for using it and still avoid any liability for fraudulent charges that do occur.