Recently I have been presented with the argument that with each new moon, the beginning of a new month begins the counting of the days so that Shabbath does not always fall on "Saturday" but seven days after the first new moon sighting. I believe that G-d set the 7 days from the beginning of time and that is the standard. Nowhere do I read in the scriptures where it states count 7 days from the first new moon and then it is Shabbat. If that were so would not G-d’s Chosen be honoring it? What do you say?

You are completely right – the days of the month begin from the new moon and the days of the week are fixed from the first Shabbat 5765 years ago.