What is the material that is there now? Needs to all be ripped apart and framed. You will need to basically frame a square opening and tie back to different length blocking to square it off. Not that tough.

@millworkman

It's stucco on top of wood and concrete as far as I can tell. I've already screened in the rest of it.

You can put a mosquito net curtain in. You'd put a curtain rod at the ceiling, a bit longer than the opening is wide. They you get at least 2x the width in mosquito netting. You can do it in two panels. You probably want a lightish metal chain at the bottom to keep it from blowing on windy days. I put a header and footer on mine of sturdier fabric.

It's very effective for bugs, although it might not be as effective if you have pets.

@Sigrid I had thought about doing this but we have pets that we want to let out on to the porch sometimes.

We have a magnetic screen at our back porch door. You walk right through and it reseals. You could mount that on the inside of the wall, ignoring the arch. It attached on our house with double stick tape and velcro. You'd have to check out the best way to attach.  

Can you add a square-framed screen door that opens outward? You might have to make your stoop bigger.

It might be helpful if we could see the entire front of your house in order to come up with some ideas for you.

Or a custom door…

@kazzh a custom door won't make much difference here.

A welding shop could fill in the opening like a fence with a door instead of a gate if that would look good with your house. If you frame for a door with dimensional lumber unless you plan on completely reframing and stuccoing the opening it could look tacky.

Oh well,…..