Speaking in Parliament on Tuesday, Weber excluded outgoing president Martin Schulz from any criticism after the Socialist group shelved the agreement just before Christmas.
But Weber, a German MEP, castigated the S&D group for ending the so-called 'grand coalition'.
Despite the criticism, he said the EPP group still remained "open" to cooperation on the presidency with the Socialists and appealed to them to "reconsider" the decision.
"I still hope that common sense can prevail," said Weber.
Under a deal signed by Schulz at the start of the current mandate, details of which were published by the EPP group on Tuesday, Weber said the EPP group was due to take the presidency for the second half of the legislature, starting this month.
Instead, the Socialists have put forward their leader, Italian deputy Gianni Pittella, for the post.
Weber said that his group had agreed "in the interest of stability" that Schulz would remain in the job, but the presidency has suddenly become vacant by Schulz's shock decision to go back to Germany.
Addressing a news conference, Weber said, "I had hoped over Christmas that the two other main groups would enter into negotiations about this but they declined so I have decided to go public."
He said the tipping point were revelations this week that Parliament's ALDE group had considered an application by Beppe Grillo, of the Eurosceptic 5 Star Movement, to join their group.
"This, for me, completely changed the picture," said Weber, who stressed that main aim of the EPP-Socialist agreement was to ensure that "extremists and Eurosceptics" in Parliament would not get a foothold in the EU decision making process.
He said that ALDE's flirtation with the 5 Star Movement and Pittella's move to cooperate with the left-wing GUE/NGL group, which he says comprises "some Eurosceptic forces," had forced the EPP's hand.
Weber said, "Parliament has always worked via partnerships and alliances. This agreement with the Socialists was not negative and has nothing to do with taking decisions behind closed doors.
"It was about cooperation between pro-European forces with the aim of minimising the influence of radicals. It is crystal clear what it meant. However, it has now been broken. I was very taken aback by this decision and the Socialists and Liberals must now bear the responsibility if the radicals in the Parliament now gain more influence.
"The EPP now appears to be the only group which refuses to enter into any sort of cooperation with Eurosceptics. The consequence is that the European elections in 2019 could result in a very unstable Parliament."
He added, "I emphasise that I exclude Schulz from any criticism. He encouraged a continuation of the grand coalition and we, in fact, voted for him to remain in post."
Weber also highlighted the credentials of the EPP group candidate, Antonio Tajani, saying he had "30 years' experience" at EU level and enjoyed the support of his group, the biggest in Parliament.