Lionel Messi opened the scoring on Saturday afternoon, before Angel Lafita made it 1-1 at Camp Nou, but Alexis Sanchez looked to have given the home side the win on 67 minutes.
However, late drama ensued, as Lafita rose highest to head home deep into injury-time to salvage a point for struggling Getafe and all but end Barca's hopes of retaining their La Liga crown.
Martino said after the game: "We paid an expensive mistake. The team had desire and invention. We played well and deserved to win. It is not a desired end and not a good season. Neither for me nor for the club. It has been a very difficult season.
"I think it is the responsibility of the coach, who leads the team. Whatever we say mathematically, we have already given everything we had to win the league. We wanted to do many more things, but we could not get to the last day with options.
Martino's downbeat tone continued throughout his post-match press conference and he admitted there were problems within the club that need resolving, casting further doubt over his future.
"The players know what is dividing the club and what is the choice," he said.
"There is still time to tell my future. I do not like to avoid responsibilities and feel that I am responsible. During the season we had some shortcomings which we haven't resolved."