How to create a grid that fits perfectly in a 1500x1500 document?

A slight variation

Make a New Guide Layout. Save with a meaningful name if you like to make it a preset.  (It will easier to use in an Action if it is a saved preset.)

Make your grid using the Line tool.  The line will snap to the grid.

I made one line; copied it three times and space them out using Move > Align.  Merged the layers, copied again and rotated 90°. Merge one more time and you have a vector grid.

What I suggest you do at this stage is open the CC Libraries panel. Make a new library with a meaningful name, and drage your grid to it.  

That grid can be dragged onto any new document with square dimensions, and dragged to fit.  CC Libraries are super useful. 

I probably do things the long way around but here's how I did it:

1. Using either the Pen Tool or Line Tool draw a vector line. Set it to the width you want.

2. With the Move Tool, select that line and with the Align Tools align it to the left of the canvas. The Align Tool has two options: align to Selection or Align To Canvas - make sure it's Align To Canvas.

3. Make a copy of that line layer. Select it and align it to the right-hand side of the canvas.

4. Make 4 more copies of the line and drag them anywhere between the two aide lines already aligned.

5. Select all the lines and use Align>Distribute Spacing Horizontally.

6. Group all the lines together and make a copy.

7. Rotate 90 degrees and align to the canvas either top or bottom to finish the grid.

It took longer to write than to do!

In addition to Bojan's method, you can also set the grid to %.

I use both methods depending.

Have you tried View > Guides > New Guide Layout > use Columns and Rows without Gutter (delete value or type 0). This will create guides which you can lock using View > Guides > Lock Guides. To show/hide guides use Ctrl + H keyboartd shortcut. I do not think you can control order of guides, they will stay on top, otherwise guides will lose purpose.

Here is free script to convert guides to vector shapes, change: var lineWidth = 6 to lets say 2 what will be shape width if you want tiny shapes or to 1 if you want them tiny as guides https://community.adobe.com/t5/photoshop-ecosystem-discussions/hey-guys-how-to-convert-ps-guides-into-paths-or-strokes/m-p/13303186#M681390 After converting to vector shapes you can control position of shapes in the Layers stack.