They can be made, but no manufacturer makes them because there is no monetary incentive to since no applications for them have been developed.
A green diode is called a "direct green" and there are a very limited number of them available, typically at over $1000/diode for less than 50mW output.
Laser diode manufacturers do not see laser pointers as a valid use for their products and do not like us using "their" diodes. That is why we do not type the names of the manufacturers (we say M!t$ub|$h! or Ka$Eo) that we harvest diodes from, they don't like it either and check the internet for where their name is mentioned.
The semiconductor material's bandgap determines the wavelength. I'm not sure if it can have multiple bandgaps, but if so, then the coatings on the facets determine which wavelength gets amplified and emitted.