Cardinal Raymond Burke, an American prelate of the Catholic Church, patron of the Sovereign Military Order of Malta and former Archbishop of St. Louis, accused the Pope of sowing 'error and confusion' with words that 'do not correspond to the constant teaching of the Church'.
Burke, an outspoken critic of Francis, was supported by Bishop Thomas Tobin of Providence and Cardinal Sean O'Malley of Boston, who said the words are not approval of 'homosexual activity'.
It appears after Francis told filmmakers that same-sex couples are 'children of God and have a right to be in a family' throughout a 2019 interview that was published this week.
Francis came out in aid of same-sex unions while serving as the Archbishop of Buenos Aires, but has not spoken out on the matter publicly after he became pope.
His comments, throughout a feature-length film named Francesco that was displayed at the Rome Film Festival on Wednesday, were seen by many as tacit approval of same-sex relationships.
Reacting to the statements, Cardinal Burke wrote: 'Such declarations generate great bewilderment and cause confusion and error among Catholic faithful, inasmuch as they are contrary to the teaching of Sacred Scripture and Sacred Tradition.
'They cause wonderment and error regarding the Church's teaching among people of good will, who sincerely wish to know what the Catholic Church teaches.
'They impose upon pastors of souls the duty of conscience to make fitting and necessary clarifications.
He added: 'The context and the occasion of such declarations make them devoid of any magisterial weight.
'They are rightly interpreted as simple private opinions of the person who made them.
Bishop Tobin was one of the first to speak out in defense of Francis, also blaming him for contradicting church teachings.
The Pope's statement clearly contradicts what has been the long-standing teaching of the Church about same-sex unions,' Tobin said.
'The Church cannot support the acceptance of objectively immoral relationships.'
And Cardinal O'Malley added his voice to the objection on Thursday, stating that Francis 'strongly and consistently teaches that marriage is between a man and woman for a lifetime and that this is God's plan for having and raising children.'
He added: 'The Pope's endorsement of civil unions is not an endorsement of homosexual activity...
However, the Pope's words also drew applause from other quarters, including the Catholic head of the UN, Antonio Guterres.
'This is a clear demonstration of a fundamental principle, which is the principle of nondiscrimination,' said Guterres in New York.
'And one of the things that has been very clear in the UN doctrine on this is that non-discrimination is also relevant in the questions of sexual orientation.
'So this decision of the Pope is, of course, extremely welcome from our perspective.'