The 28-nation bloc also called for an investigation to identify those responsible for the violence, as opponents vowed to hold fresh rallies to oust President Nicolas Maduro.
"We call on all Venezuelans to come together to de- escalate the situation and find democratic solutions in the framework of the constitution," EU foreign affairs spokeswoman Nabila Massrali told AFP.
"Only peaceful and constructive engagement can halt the deterioration of the situation in Venezuela and build better prospects for its people."
Massrali added: "The acts of violence during the demonstrations in Venezuela on 19 April are highly regrettable. We are saddened by the deaths of a man and a woman and concerned at the reports of numerous persons injured."
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The statement did not mention the soldier.
"All concerned, including members of the security forces, have a responsibility to act in full compliance with the rule of law and human rights," the spokeswoman said.
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