Is it possible to drive a 2 X 16 LCD display using logic gates? (Without computer)
It is possible to make a circuit out of logic gates that would be able to initialize and display characters onto a 16x2 LCD character mode module. It is also possible to make a circuit out of logic gates that could take the output of a standard matrix keypad or even a PS2 type keyboard and route that keyboard data to the LCD module.

The big thing to think about though is that in just asking the question you imply that you are at the beginning of the learning curve of how to go about doing a project like this. I suggest that if you want to build something out of logic gates that you start with something a lot simpler till you get the idea of what is involved in putting together things to make this type if thing. Only then will you begin to be able to answer your own question as to why it may not make much sense to connect a keypad and an LCD module with dedicated logic gates. It would take a good amount of logic gates to achieve this and certainly a lot of time and effort to come to a good result. Once you had it all completed you would almost immediately begin thinking about how you could make the display show a string of characters and possibly even have ability to "edit" the displayed string. Now the logic gate implementation becomes a nightmare of a whole new project.

So there is a reason that the whole world has transitioned to doing things like this with a microcontroller. Very simple interfaces are used to connect things like matrix keypads and 16x2 LCD modules to the GPIO interfaces of the microcontroller. These simple interfaces are easy to build and test to make sure that they work. Once in place it becomes a simple software job to provide the logical connectivity between the keypad and the LCD module. And that software can be as simple as the original design scenario that you asked about to just take a single input from the keypad and show it on the display or it can be adapted to show almost anything on the display ... for example showing a count if how many times a key has been pressed ... or show how long a key has been pressed ... or show a whole user interface that operates your device.