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