Will the Boeing 737 be built with a geared turbofan engine?
The current 737 NG has only one engine choice, the CFM-56. Although this means less choice for the customer, it allows the manufacturer to focus on supporting only one type of engine. The comparable A320 has options for the CFM-56 and the V2500.

The A320neo effectively replaced the CFM-56 with the LEAP-1 and the V2500 with the PW1000G. The 737 MAX will replace the CFM-56 with the LEAP-1, and continue to provide only one engine choice. Note that on the A320neo both engines offer comparable performance overall, and orders are basically split between the two.

Although the 737 MAX is behind the A320neo in sales, it's unlikely that adding another engine choice would make much difference for the considerable expense it would require. The cheapest option is to use the PW1000G, and that still requires a lot of work for the manufacturer to integrate it into the airplane, and then test and certify the 737 with that engine. As with the LEAP-1 and CFM-56, Boeing would prefer to have their own minor version of the engine certified, which adds more expense. There's just no incentive to add another engine option at this point.

A future version of the 737 could possibly use a geared turbofan, but it's too early to tell if such a version will even be built. It certainly remains an option for future designs as engine manufacturers develop the technology further.