The Croatia captain, making his 205th appearance for his country, was unable to prevent a humiliating night in Rijeka as England produced one of their most emphatic victories.
Harry Kane and Bukayo Saka both scored twice, while Jude Bellingham, Anthony Gordon and Morgan Rogers were also on target as England tore apart the hosts.
The result represented Croatia's heaviest defeat since independence in 1991 and left Modric deeply critical of his team's performance.
"It is difficult to find an explanation for such a performance without duels, fouls, aggressiveness and togetherness," the 41-year-old said.
"I've been here for 20 years, but for the first time I feel ashamed after a game.
"You can lose, or play against an opponent who is better - and they are right now. But losing like this and ruining the reputation we've built over the years is unacceptable."
England were already in control inside eight minutes when Kane curled a superb finish into the top corner after controlling Bellingham's pass.
Bellingham then doubled the advantage before Gordon headed in a cross from Jarell Quansah, with Saka adding a fourth before the interval.
Kane struck again shortly after half-time and Saka completed his brace in the 68th minute, before Rogers rounded off the rout.
Modric was substituted in the 57th minute, meaning he could only watch from the bench as England added the final two goals.
The defeat came just days after Croatia had been beaten 4-1 by Spain, piling further pressure on coach Slaven Bilic during his second spell in charge.
Bilic accepted responsibility for the collapse but said there was no single explanation for what had happened.
"Since taking over this summer, there have been issues - there certainly have. But the responsibility is mine," he said.
"It was simply a collective blackout."
Croatia's latest setback is a stark contrast to the country's previous meetings with England. Modric played a key role when Croatia eliminated the Three Lions in the 2018 World Cup semi-finals before going on to finish runners-up, while he also helped them reach third place at the 2022 World Cup.
Now facing Spain again on Tuesday, Modric called for a response rather than further blame.
"We need to come together and stay together because another difficult game awaits us," he said.
"We have to show our true colours - that we have fight and pride, desire, willpower and unity. You can't lose that overnight."
Luis de la Fuente believes Lamine Yamal is already on the road to becoming a Spain legend after the teenager continued his remarkable international form in Saturday's 3-1 Nations League victory over Czechia.
Bruno Fernandes has called for unity in Portugal amid the controversy surrounding Cristiano Ronaldo's departure from the national team camp, insisting the veteran remains the country's "greatest symbol".
Roy Keane believes Real Madrid and England midfielder Jude Bellingham has all the qualities to go on to become a legend of the game.
England produced their most emphatic display of the Thomas Tuchel era as they demolished Croatia 7-0 in their UEFA Nations League clash in Rijeka.
