Speaking to the BBC, Varney said he was "totally devastated" by the incident and said his "heart goes out" to those affected.
Media interest has been intense since two carriages crashed on The Smiler ride at Alton Towers, leaving passengers trapped for hours while four suffered leg injuries.
Varney said: "Our business is about giving people safe fun memorable experiences, with the emphasis on safety. And yesterday something dreadful happened."
However, in the same interview Varney said there had been an "awful lot of misreporting" about alleged previous problems on the Smiler. He said there were "teething problems" but these didn’t relate to customer safety and were not connected to the apparent problems with breaking locks in the recent case.
He also defended the group against complaints of a slow response on the ground, saying there was "no laxness" to it and the team are "well trained".
A statement from Varney, issued by Merlin, said: "This has been a terrible incident and a devastating day for everyone here. We have a very strong record of safe operation of our rides here at Alton Towers and it is our priority. I would like to express my sincerest regret and apology to everyone who suffered injury and distress today and to their families.
"I would like to thank the emergency services and my own teams for their swift and effective response yesterday – they were outstanding. A full investigation is now underway and we will continue to work closely with the emergency services and the Health and Safety Executive to better understand the cause of this terrible accident."
The decision was taken to close the park for the time being, which has been publicised via a series of tweets from the Alton Towers Twitter account:
1of4: The decision has been taken that Alton Towers theme park will remain closed tomorrow 4/6/15 following the incident on The Smiler— Alton Towers Resort (@altontowers) June 3, 2015
2of4: Guests with pre-booked tickets & those arriving at the theme park will have the choice of a full refund or an alternative dated ticket— Alton Towers Resort (@altontowers) June 3, 2015
3of4: Guests with pre-booked tickets have the option to use their tickets tomorrow at an alternative attraction. See website for more info— Alton Towers Resort (@altontowers) June 3, 2015
4of4: We will be posting all updates on the theme park closure on both our website and social media channels— Alton Towers Resort (@altontowers) June 3, 2015
Nobody in the comms team at Merlin or Alton Towers was available to talk to PRWeek at the time of publication. Merlin’s agency roster includes The Firm for corporate comms and Tulchan for financial PR, both of which declined to comment.