The majority of 180 bishops from nine Amazonian countries also called for the Vatican to reopen a debate on ordaining women as deacons, saying: "It is urgent for the church in the Amazon to promote and confer ministries for men and women in an equitable manner."
The proposals were contained in a final document approved on Saturday at the end of a three-week synod on the Amazon, which Pope Francis called in 2017 to focus attention on saving the rainforest and better ministering to its indigenous people.
The Catholic Church, which contains nearly two dozen different rites, already allows married priests in Eastern Rite churches and in cases where married Anglican priests have converted.
However, if Francis accepts the proposal, it would mark a first for the Latin Rite church in a millennium.
The celibate priesthood has been a tradition of the Latin Rite Catholic Church since the 11th century, imposed in part for financial reasons to ensure that priests' assets pass to the church, not to heirs.
Francis told the bishops at the end of the voting that he would indeed reopen the work of a 2016 commission that studied the issue of women deacons. And he said he planned to take the bishops' overall recommendations and prepare a document of his own before the end of the year.
Some conservatives and traditionalists have warned that any papal opening to married priests or women deacons would lead the church to ruin. They accused the synod organisers and even the pope himself of heresy for even considering flexibility on mandatory priestly celibacy.
They vented their outrage most visibly this week when thieves stole three indigenous statues featuring a naked pregnant woman from a Vatican-area church and tossed them into the Tiber River.
The statues, which conservatives said were pagan idols, were recovered unscathed by Italy's Carabinieri police.
They were on display on Saturday as the synod bishops voted on the final document, which was approved with each paragraph receiving the required two-thirds majority.
The most controversial proposals at the synod concerned whether to allow married men to be ordained priests, to address a priest shortage that has meant some of the most isolated Amazonian communities go months without a proper Mass.
The paragraph containing the proposal was the most contested in the voting, but received the required majority at 128-41.
The proposal calls for the establishment of criteria "to ordain priests suitable and esteemed men of the community, who have had a fruitful permanent diaconate and receive an adequate formation for the priesthood, having a legitimately constituted and stable family, to sustain the life of the Christian community through the preaching of the Word and the celebration of the sacraments in the most remote areas of the Amazon region".
The paragraph ended by noting that some participants wanted a more "universal approach" to the proposal - suggesting support for married priests elsewhere in the world.
Francis has long said he appreciates the discipline and the gift of celibacy, but that it can change, given that it is discipline and tradition, not doctrine.
History's first Latin American pope has been particularly attentive to the argument in favour of ordaining "viri probati" - or married men of proven virtue - in the Amazon, where Protestant and evangelical churches are wooing away Catholics in the absence of vibrant Catholic communities where the Eucharist can be regularly celebrated.
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