Which week day(s) cannot be the first day of a century?

In four hundred years there $97$ leap years and $303$ non leap years. This contributes a total of $97\cdot 2+303=497$ 'free days'. This is a multiple of $7$, which means that the weekday cycle repeats every $400$ years. Thus only four weekdays (out of seven) will ever be first days of a century.

$400$ years together have $400\cdot365+97=146\,097$ days, which is divisible by $7$. Therefore everything repeats after $400$ years. According to Wolfram Alpha January $01$ of the years $2000$, $2100$, $2200$, and $2300$ are a Saturday, Friday, Wednesday, and Monday, respectively. It follows that no century begins with a Tuesday, Thursday, or Sunday.

Letting the centuries begin on January $01$ of year $100k+1$, the missing days would be Wednesday, Friday, and Sunday.

The leap year formula is not perfect. More precisely a year is 365 days, 5 hours, 48 minutes and 46 seconds long. On the average year of the Gregorian calendar, we fall behind 27 seconds. So every 3236 years or so, we will have to skip a leap year to remain on target. Due to this, every day will be possible to start a century.

History has shown on multiple occasions that we will tweak our calendar to maintain the familiar seasons. It's also easier to change one day than to shift the equinoxes, solstices, crop planting dates, school opening/closings, etc. So even though this is a hypothetical situation, I 100% guarantee the leap years will be adjusted before July is allowed to become chilly on the Northern Hemisphere!