Fomenko's side finished with no points and bottom of Group C having failed to score a single goal in their three matches.
They lost 2-0 to Germany in their opening game and were then beaten by Northern Ireland by the same score line, before their campaign ended with a 1-0 defeat to Poland.
"I have two main reasons for our failure. First of all, we were not united as a team," Fomenko said on Ukrainian Football Federation website.
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"Second, some players were not ready in terms of psychology. But I want to stress that I am not blaming the players. All responsibility is on me.
"I did not find the way into their mind. Probably, they did not fully realise the scope of the tournament. I should have held more personal conversations."
Ukraine's preparations for the tournament suffered a setback after an unseemly row between two of their high-profile players. Shakhtar mid fielder Taras Stepanenko and Dynamo Kiev's Andriy Yarmolenko were involved in a spat during a league game.
Fomenko had said on Tuesday that their final game at the Euros against Poland would be his last as coach.
"There is no sense to carry on cooperation after we failed to fulfil our goal at Euros," he added on Thursday. "I take full responsibility for the poor results and quit the post.
"I would like to apologise to the Ukrainian supporters because I did not justify their expectations."
Former striker and assistant coach Andriy Shevchenko is a possible successor, according to media reports.
(Reporting by Igor Nitsak; Editing by Toby Davis)
