Is there a way to move a split page to a new tab in Vim?

Very easily, use CTRL+W, SHIFT+T.

See the help page:

:help CTRL-W_T

Note that this is case sensitive: <C-W>T is different to <C-W>t.

The long-and-straight-forward way would be to open a new tab and open the file's buffer there.

:tabnew
:b FILE_NAME

:b can TAB-complete from arbitrary parts of the file name, so this shouldn't take too long.