If I were to approach a gallery here are the things I would think about...

To a gallery I would say:

This is just an introduction. I just want to make you aware of my work, please feel no obligation to respond

Here’s who I am

Here’s what I create

Here’s why I think I would be a good fit for your gallery

Here’s how I will help you sell my work

I might include a quick -- Here’s what I’m passionate about these days

I would ask, how can I make this email fun, concise, and I'd attach one to three jpegs of my best art and maybe a good photo of me as well

I'd have an easy link to my well put together website full of good photographs of my best work

Really make sure you answer: Why should Gallery X want to work with you?

Really make sure you answer: How will you help Gallery X sell your work?

I would include my phone number