“People have been saying ‘I hope your children die’. I’d like people to put themselves in my shoes and think what it’s like to get threats to my family,” Moarata told the Cadena Cope station.
“My wife and children have come to the stadium in Seville with Morata on the back of their shirts and people have been shouting at them.
“It’s complicated. I understand people booing me for missing chances but there’s a limit,” added Morata.
Morata, who is on loan at Juventus from Atletico Madrid, has missed a number of chances in the tournament, including a penalty in his side’s 5-0 win over Slovakia on Wednesday.
The 28-year-old said thay he had a sleepless night after missing the penalty against Slovakia, but does not regret taking the spot-kick.
Morata said that despite the abuse and criticism he has faced he feels a huge level of support from his team mates, adding former Spain great Iker Casillas had called him to lift his spirits.
“I’m proud of the fact I picked up the ball (to take the penalty) after people booed me in the warm up. A few years ago I would have been devastated but I’m really motivated. Whoever thinks the opposite doesn’t know me,” said Morata.