In a statement on Thursday, Mr Cruz said that he planned the trip for his daughters, “wanting to be a good dad”.
Photos of the Republican lawmaker at an airport emerged on Twitter on Wednesday, fuelling reports that he had left Texas amid the energy crisis.
Texas has seen power and water outages due to the uncommonly frigid weather.
In a written statement released on Thursday afternoon, Mr Cruz described an “infuriating week for Texans”.
With schools closed in the state, the senator said he booked the vacation for his young daughters who “asked to take a trip with friends”.
Mr Cruz did not apologise for making the trip amid the state of emergency in Texas, but said that he and his staff “are in constant communication with state and local leaders to get to the bottom of what happened in Texas.”
The Republican will be returning to Texas on Thursday afternoon, according to his statement. It is not clear if his return flight was originally scheduled for Thursday or is a change in plans.
The Houston Police Department told CBS News that a member of Mr Cruz’s staff contacted authorities on Wednesday to request assistance upon Mr Cruz’s arrival at Houston’s international airport.
The senator arrived at the airport on Wednesday, Chief Art Acevedo said, and officers “monitored his movements through the terminal”.
What’s the situation in Texas?
The news comes as hundreds of thousands of Mr Cruz’s fellow Texans woke up to a fourth day without power, after extreme winter weather overwhelmed the state’s energy grid.
At least 24 people have died amid the winter storms. Outages are expected to continue for days.
Texas officials have also ordered seven million people in the state to boil tap water before consuming it due to damaged infrastructure and pipes.
And with the sustained frigid temperatures, frozen pipes have caused a drop in water pressure in homes and hospitals. Governor Greg Abbott has now asked residents to shut off water to their home if possible to help keep pressure up.
Mr Cruz had been active on Twitter in recent days, sharing updates about power in the state.
In his Thursday statement, Mr Cruz said that his family had also lost heat and power, just like other Texans.
“The greatest state in the greatest country in the world has been without power,” he said.
Who is Ted Cruz?
Mr Cruz is a high-profile member of the Republican party.
As a federal lawmaker, Mr Cruz does not have a direct role in Texas’ emergency response, but residents often reach out to their elected officials – like Mr Cruz – during natural disasters for help accessing resources.
He ran unsuccessfully for president in 2016. An ally to former president Donald Trump, the senator was present in Washington last week for Mr Trump’s second impeachment trial, where he voted to acquit.
What’s the reaction been?
The senator’s travel plans amid the crisis in Texas drew immediate criticism.
“People in Texas are literally freezing to death and yet Ted Cruz went on vacation to Cancun,” wrote Democratic Congresswoman Pramila Jayapal.
Others noted how the added scrutiny may hurt Mr Cruz comes as he reportedly contemplates a second presidential bid in 2024. The lawmaker’s current Senate term will expire in early 2025.
And some took a more humorous take on the senator’s southern vacation – posting photos of fake missing person’s signs of the Texas lawmaker.
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But some conservatives came to Mr Cruz’s defence – author and activist Brigitte Gabriel wrote: “Senator Ted Cruz is one of the hardest working men in the country, he deserves a vacation.”