VATICAN CITY—A former Vatican diplomat admitted on Friday to downloading and sharing child pornography images on Tumblr, but suggested that his culpability was mitigated by a troubled state of mind.

Msgr. Carlo Alberto Capella, who was arrested in April for allegedly violating Vatican City law against possession and dissemination of child pornography, told a three-judge panel on the first day of his criminal trial that he didn’t deny the charges against him.

At...