The 52-year-old was heading for the locker room at half-time with his team in charge of their clash against Indianapolis when he suffered what the Texans have described as a mini-stroke.
It has been confirmed that Kubiak "experienced a transient ischemic attack", and doctors at the Houston Methodist Hospital expect him to make a full recovery.
But neither Kubiak or the Texans are able to predict when he will be able to resume his coaching duties.
Defensive co-ordinator Wade Phillips took over for the second half on Sunday, when the Colts overturned an 18-point deficit to snatch a 27-24 victory.
Kubiak issued a statement on the Texans website, which read: "I want to thank my family, the McNair family, the Texans organization, the doctors and staff at Houston Methodist Hospital and the entire Houston community for all the love and support we have received over the past three days.
"I've been through an ordeal and my focus now is to get back to good health.
"Doctors have told me I will make a full recovery but we have not determined when I will be cleared to return to the office. Again, thank you for the support and concern."