The abuse aimed at Alvaro Morata and his family by some Spain supporters crossed the line and is "not acceptable", according to team-mate Dani Olmo.
Morata has been the focus of attention for La Roja since being jeered by his own fans during a pre-Euro 2020 friendly with Portugal after missing several opportunities.
He was then criticised for his displays in Spain's 0-0 draw with Sweden in their Group E opener and 1-1 draw with Poland, a game in which he scored but missed more chances.
The Juventus striker, whose loan from Atletico Madrid was extended for another season last week, then missed a penalty in Wednesday's 5-0 win against Slovakia.
Morata revealed in an interview that he has received vicious messages on social media during the tournament, while his wife and children have also been targeted in public.
"I would like people to put themselves in my shoes and think what it's like to get threats towards my family, people saying: 'I hope your children die'," he told Cadena Cope.
"I've had to leave my phone outside my room. 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."
Morata has had five big chances at Euro 2020 so far. Only Cristiano Ronaldo (six) has had more, though the Portugal skipper has scored five times.
The former Chelsea striker's four Opta-defined big chances missed is the most of any player in the tournament, followed by Gerard Moreno and seven others on three misses.
Spain boss Luis Enrique has repeatedly leapt to the defence of Morata for his performances and is expected to stick with the 28-year-old for the last-16 tie with Croatia.
And team-mate Olmo, who started two of the group games, feels the vitriol Morata has faced has gone too far.
"Myself and all the team are against the threats," he said at a news conference on Saturday previewing Spain's knockout clash with Croatia on Monday.
"It is normal that we are criticised, it is part of our sport, football, and we are exposed to it. We accept it. But going beyond it is not acceptable.
"I think people should put themselves in the skin of the player. We all are human being and we need to respect it.
"Again, I understand if we are criticised as it is part of our profession, but going beyond it is too much."
After winning Euro 2008, World Cup 2010 and Euro 2012, Spain have since been eliminated from both of their last two major tournament knockout-round matches.
Croatia won the most recent of their meetings with Spain 3-2 in November 2018, a year before Olmo earned his first senior cap for La Roja.
Olmo spent his formative years in Dinamo Zagreb's academy and could have represented Croatia, but he instead pledged allegiance to Spain.
"It's true there was an interest from Croatia, but I believe I have always made clear my side," he said.
"I was thankful for that interest as I have special love for Croatia, but I always said my dream was to be here with Spain. At the end, I got it, and I am where I always wanted to be.
"Croatian people have showed to be very competitive in all sports, not only in football.
"It is a country not as big as Spain, France or Germany, but they compete even more in every single sport. And they are very patriotic. They give their all for their country."
Asked if head coach Luis Enrique has asked for any inside information on Croatia, Olmo said: "Not yet! I think he knows their team very well.
"I also obviously know them very well and have played with many of them. If he asks me something, I will give him my opinion, as usual.
"I've spoken to a couple of their players already and cannot wait to play the game."