Lamara Bell, 25, was critically injured in the crash off the M9 near Stirling on Sunday July 5 and died in hospital yesterday, while her partner, John Yuill, 28, died at the scene.
They were only found in the car on Wednesday, around 72 hours after the accident, despite the incident having been reported to police on the day it happened.
Police Scotland Chief Constable Sir Stephen House said he believes it is right for him to stay and provide leadership within the organisation.
He said: “In anything like this I consider my position. I think you would be inhuman if you didn’t. You see what’s happened over the last week and think about that, and I certainly have.
“I believe the right thing to do is to stay to get through this process, to get through this tragic event and the series of events that followed it and to see what can be done to fix the situation.
“I don’t want anyone out there thinking I’m the type of person that says ‘I’ll never go, I’ll have to be forced out’.
“If I come to the conclusion that I should resign then I will resign. I don’t believe that’s the case at this moment in time.
“This organisation needs leadership, I’m providing that leadership.”
Sir Stephen last week apologised to the relatives of the two crash victims and admitted police had “failed both families”.
He said today the force is in the middle of “massive change” in how it handles calls which still has some way to go.
“I remain confident and convinced the reforms we’re pushing through is the right way to go and provides a more efficient and more professional service,” he said.
“The tragedy is that I’m saying this against the background of two people who have died and that’s been our error which we’ve acknowledged.
Scottish Justice Secretary Michael Matheson has ordered Her Majesty’s Inspectorate of Constabulary in Scotland (HMICS) to undertake an urgent review of all police call handling following the incident.
The case is also subject to an independent probe by the police investigations and review commissioner (Pirc).
Asked whether he still has full confidence in Sir Stephen, Mr Matheson told BBC Radio Scotland: “Yes, he still has my confidence.”
He added: “I think the chief constable has got a number of challenges which he has dealt with since coming into the post, and with the merging of our eight previous legacy forces and creating Police Scotland.
“He has acknowledged that there have been mistakes made over the last couple of years including a mistake made in very recent times with this particular case, and he sought to try and address them, so I think it’s important that the focus now has to be on making sure that this type of thing does not occur again and we learn the lessons from that.”
Mr Matheson said there was no indication of any “systemic failure” in the police call handling process.
Scottish Labour justice spokeswoman Elaine Murray had said earlier that Sir Stephen should be considering his position, while acting Scottish Labour leader Iain Gray said the Justice Secretary is “the only person in Scotland who still has confidence in the chief constable”.
The couple were reported missing after visiting Loch Earn, Stirlingshire, in a blue Renault Clio.
It emerged that a member of the public contacted Police Scotland at around 11.30am on Sunday July 5 to report that they could see the vehicle down an embankment near the M9 slip road at Bannockburn.
The call was answered, but ‘’for reasons yet to be established’’ it was not entered on to the police systems and no action was taken at the time.
The car was found when officers were called to the scene by another member of the public on Wednesday.
Meanwhile, Ms Bell’s brother Martin has posted his thanks on Facebook to the community in Falkirk.
He wrote: “I’m blown away by the effort and time our community in falkirk are putting into making the loss of my sister as painless as possible for my family.
“There has been two donation pages been set up one for my family and johns family for any funeral arrangements ...and another where any money raised will go to the kids that have been left behind with out a parent (sic).
“There has also been this fund raising night set up by Camelon Juniors Fc.”