Can I secure a gas line to an exterior wall?

I have a black iron pipe gas line that comes down my unfinished basement wall to supply the dryer. It is not supported on this vertical section so it rocks back and forth. Can I clamp it to the brick wall?

Sounds like a gas line I had going to the water heater. The clamps holding it in place came loose and it dangled moving about especially when someone came in and out of the basement door from the garage. I could had clamped it back down but instead I had the lines moved, I had the space in which I was able to.

Oh, wait Ian, I just read where you said, an exterior wall. That is a horse of a different color.

Yes Ian, you can and should.

What I use when I need to fasten anything to concrete is a small rotary hammer/drill, a 5/16" SDS bit, and 3/8" lead sleeves. After drilling a hole, I tap a sleeve into it and use a #12 sheet metal screw to secure whatever needs to be tied down. It's quick, easy, and inexpensive.

I would use a 3/8th ceiling flange and a microfix pipe clamp along with some smack pins to secure the ceiling flange to the walls.

Very secure and will last forever.

I always used a trigger operated .22 shooter and 1.25" masonry nails when I needed to attach something to a masonry wall.

If there is "wiggle room" behind the pipe, I like a block of wood , or even just a wood shim, to keep the pipe from rubbing on the stucco

They make clamp support plates especially for this. The ones I have on my concrete wall (you could attach them to brick the same way) have two screw holes. In the middle between the two screws is a threaded hole that you use threaded rod to attach the clamp. the advantage of this threaded rod is that you can make the length anything you need to keep the pipe where you want it, and not put stress on the joints. You could use TapCon screws or lead anchors or probably even plastic anchors in the brick to hold it in place. On brick, you may not want to use a powder activated fastener as it may crack the brick. Most brick would drill easily with a carbide bit of the proper size. A TapCon would result in the smallest hole, but you have to buy their drill bit to ensure proper engagement. If you have a hammerdrill, it is fast work. Otherwise, it often takes a bit longer with a normal drill. It's easy to overheat the bit with a normal drill.

The real question is "why wouldn't you and/or what would prevent you from doing it?"