The 22-year-old limped off in the 63rd minute of his team’s 6-1 Premier League defeat at Bournemouth on Saturday and tests revealed bad news.
Robertson will now also miss Scotland’s World Cup qualifier against England at Wembley on November 11.
A Hull statement read: “Andy Robertson is likely to be sidelined for up to eight weeks with the injury sustained against Bournemouth at the weekend.
“The Scottish international damaged his calf during the second half of the defeat at the Vitality Stadium, with subsequent scans showing that the left-back is set for a spell on the sidelines.
“Robertson will miss several crucial Premier League games which could define his team’s season, with Hull facing Stoke, Watford, Southampton, Sunderland, West Brom, Middlesbrough and Crystal Palace inside the next two months.
He wrote on instagram: “Looks like I’ll be a supporter for the next few weeks! Hopefully the lads can kick on and I’ll be back in no time.”
The former Queen’s Park and Dundee United player was Scotland’s most impressive performer during their 1-1 draw with Lithuania the previous Saturday but made way for Celtic’s Kieran Tierney in their subsequent defeat by Slovakia.