The airport posted on Twitter explaining that all commercial flight operations had been stopped and that certain roads leading to it were closed due to flooding.
A record 75cm of rain has already fallen on Houston in the wake of the hurricane over the weekend, and forecasters say the total rainfall could nearly double later this week.
BBC News reports that more than 3,000 people have been rescued in and around the city and at least nine people are reported to have died in incidents related to the storm, Texan officials say.
A spokesperson for the carrier said: “Clearly we are affected in the way that everyone that flies from Houston has been affected.”
Houston’s George Bush Intercontinental airport is a major hub for United Airlines.
The carrier has cancelled all flights from “until at least noon on Thursday August 31”.
It added that flights to and from Texas and the Gulf Coast may also be affected.
The carrier said “severe weather and unsafe road conditions near the airport” meant it was unable to operate.
However, three United flights brought in critical supplies over the weekend to help with the relief effort.
British Airways has also cancelled its Houston flights.
A spokesperson said: “As a result of the closure of Houston’s main airport we have cancelled our flights up to and including Wednesday August 30.
“We are doing what we can to assist customers who have been caught up in the storm or who wish to change their travel plans as a result of the flooding.
“The safety of our customers and crews are always our top priorities and we are keeping our flight schedules later this week under review as the storm and flooding moves through the area.”
Houston’s William Hobby airport, which mainly offers domestic flights, also remains closed.
Elsewhere, passengers onboard three Carnival Cruise Line ships and a Royal Caribbean International ship were rerouted.
Passengers were reportedly left “stranded at sea” with Carnival ships Valor and Freedom unable to dock at Texas’ Port of Galveston.
The Carnival ships had been set to arrive in Galveston on Saturday but were instead detoured to New Orleans according to The New York Post.
Both ships docked in New Orleans to bring on new supplies and give passengers the chance to disembark.
Passengers were also given the option to stay onboard the ships until they could return to Galveston on Tuesday “at best”, according to port officials.
Carnival’s Breeze was also set to arrive in Galveston on Sunday.
However, it also took the New Orleans detour, while Royal Caribbean’s Liberty of the Seas headed to Miami and is expected to return to Galveston on Friday, four days late.
Carnival Cruise Line said in a statement: “The port announced it would be closed for at least the next 24 hours, so our earliest potential opportunity to debark current guests is on Tuesday.
"As you can imagine, this is a fluid situation, and we have no way of knowing if the port opening could be delayed further.
"We will continue to maintain close contact with port officials in Galveston monitoring conditions on Galveston Island, and within neighbouring areas including Houston, and will provide ongoing updates as we receive additional information."
