Britain's Raducanu, the 2021 US Open champion, asked for medical advice five games into the match before bowing out after losing the first set 6-1 in Seoul.
The Brit had treatment on the same foot during her last-16 win over China's Yuan Yue on Thursday.
Raducanu, ranked 70th in the world, has struggled with injuries since her stunning US Open victory, and has failed to build on her success.
She was looking to make an impact at the Korea Open, which was hit by a string of last-minute withdrawals, including by world number one Iga Swiatek, US Open runner-up Jessica Pegula and former Wimbledon champion Elena Rybakina.
Raducanu had her serve broken four times by Russia's world number 13 Kasatkina, whom she has never beaten in two previous meetings.
"I feel sorry for Emma for retiring during the match," said Kasatkina, who will play Russian number four seed Diana Shnaider or Ukrainian number five seed Marta Kostyuk in the semi-finals.
"It's a pity to get injured, especially in the deeper stages of the tournament. I wish her a speedy recovery and get ready for the next tournament."
All four quarter-finals were taking place a day later than scheduled after all play was washed out on Friday.
Brazil's number three seed Beatriz Haddad Maia booked her place in the semi-finals with a 6-2, 6-1 win over Russia's Polina Kudermetova.
Haddad Maia will play Veronika Kudermetova, who beat Bulgaria's Viktoriya Tomova 7-5, 6-3.
The semi-finals were set to be played later on Saturday.
