LONDON — Bishop Justin Welby, the new archbishop of Canterbury and spiritual head of the world’s estimated 77 million Anglicans, pledged Friday to seek reconciliation in some of the most contentious issues of gender and sexuality that have split the Anglican Communion.

Soon after Prime Minister David Cameron announced his appointment, Bishop Welby, 56, a former oil company executive, made it clear that he endorsed earlier church statements criticizing government plans to legalize same-sex marriage.