If you change your data, formatting the dates as a number with no decimal places, the chart will change the axis labels to those numbers excluding the week-end dates (note that dates are held internally as the number of days since 01/01/1900). You can then format the chart axis to display those 'numbers' as dates and the week-end dates will not re-appear.
If it is important that your data table must have dates then create another column with those dates formatted as a number and change the font to 'white' so that they cannot be seen.