Anthony Martial's goal deep into injury time gave United a 2-1 victory at Wembley to end Martinez's hopes of bringing Everton their first trophy since 1995's FA Cup win.
Asked whether he felt he was still the man to take Everton forward amid growing speculation that this was a game he had to win to survive, he said: "I am. I have got so attached to this football club and my ambition is exactly the same as this club's ambition.
"We have handled our assets really well, the squad has a great valuation. We develop young players very well and I share the drive the football club has, which is to get into the Champions League and to bring silverware back to Everton."
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Martinez has faced growing unrest from Everton's fans, who jeered their team off at half-time at Wembley before a second-half rally brought them back into the game.
There will now be renewed pressure on Martinez after the end of their trophy hopes and with Everton lying in 11th place in the Premier League.
He said: "I just feel that every season we are developing and showing certain signs that we can please our fans and bring them the silverware they deserve but of course this is a difficult day."
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Relive Manchester United's win over Everton
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Martinez defended striker Romelu Lukaku after he endured a miserable semi-final of missed chances and had a second-half penalty saved by Manchester United goalkeeper David de Gea.
He said: "I always feel strikers need to be measured in the chances they create and sometimes you need to give credit to the goalkeeper. I thought we created as a team and he showed his power and his uniqueness. I don't think that's a worry at all.
"I thought his second-half performance was one of a mature footballer. He held the ball up and we played around him. He is only 22. It shows we want to give young players big roles."
"The feeling we share now is that this is as cruel as football can be," said Martinez.
"I thought it was a terrific game but it took us 45 minutes to get into our rhythm, although then we showed a strong mentality to stay in the game and we created the best two chances.
"In the second half, I thought we built up incredible tempo and intensity to impose ourselves on the game but at the end the whole story becomes very heartbreaking."

The 20-year-old made his Warriors debut in the win against Leigh and his start at Wolves was just his second outing.
He scored 20 tries in 22 games on loan at Championship side Swinton in 2016 on a dual registration deal, while also working as a trainee accountant.
"I'm really, really pleased for Liam," Wane told BBC Radio 5 live.
"His effort in training has been fantastic. With the pre-season he's had, he earned the right to play.
"I thought they were four quality tries and quality finishes.
"It is a shot in the arm for our youth system as it shows to everybody that a first-team start is never that far away. You just take your chances when they come and he's taken his."
Wane also confirmed that England forward John Bateman is expected to be out for three months with a shoulder injury.

Jennifer Fitzgerald's ex-boyfriend left the car at O'Hare airport's employee car park. She settled with the city out of court for $4,500.
The vehicle accumulated 678 tickets before it was finally towed last year.
Ms Fitzgerald said she had been unable to gain access to the car park, according to court records.
She continued to receive tickets even after the state of Illinois revoked the car's licence plates.
Ms Fitzgerald sued the city to clear the tickets and penalties.
Her lawyer, Robin Omahana, told DNAinfo Chicago the city had disregarded its own towing rules.
"They had a little egg on their face with writing so many tickets on one car," Mr Omahana said.
City law department spokesman Roderick Drew told the Chicago Tribune that the city had tried to settle with Ms Fitzgerald before the case went to court but she declined.
Under the terms of the settlement, her ex-boyfriend Brandon Preveau will reimburse her $1,600, with Ms Fitzgerald paying the rest on a monthly plan.
Everton manager Roberto Martinez insists he is still the right man for the job after an FA Cup semi-final defeat to Manchester United increased pressure on his position.
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Wigan Warriors head coach Shaun Wane has praised the finishing of Liam Marshall after he scored four tries in Thursday's 38-16 win at Warrington.
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A Chicago woman has settled $105,000 (£67,540) in parking fines amassed after her car was abandoned at an airport in 2009, local media report.