This may be new since the question, but (LibreOffice 188.8.131.52 on Linux), Edit->Fill->Series. Choose Date as the series type and Weekday as the Time unit. I found that, after the fill, I had to select the filled cells and Format->Cells to select date formatting. (The original fill was a series of numbers.)
EDIT: Step-by-step instructions, as requested, this is now on 184.108.40.206, although I'm sorry I can't help with Windows:
- Select the range you want to fill. Start with it completely empty.
- Edit menu, Fill, Series
- Select "Down" under Direction, "Date" under Series type, and "Weekday" under Time unit.
- Enter the starting date in Start value (I used YYYY-MM-DD). End value blank, Increment 1.
- Click OK.
- Cells are now filled with numbers and selected.
- Format menu, Cells...
- Numbers tab, "Date" under category, choose whatever you want under "Format", click OK.