What does Acts 10:34 mean?

Peter is in Caesarea Maritima, the Roman capital of Samaria and Judea, in the home of a Roman centurion, Cornelius. Peter is facing a crowd of Cornelius' soldiers, family, friends, and household members, as well as six Jesus-followers from Joppa (Acts 11:12). Two day before, he'd had a strange vision wherein God repealed the kosher food laws (Acts 10:9–16). Moments before, Cornelius related the story that an angel told Cornelius—a Gentile—to listen to what Peter had to say (Acts 10:30–33).

Peter had seen Jesus care for a Syrophoenician woman (Mark 7:24–30), speak kindly with a Samaritan woman (John 4:1–45), and even praise a trusting centurion (Matthew 8:5–13). Peter had even watched the Holy Spirit fall on a town full of Samaritans (Acts 8:14–17). But the Syrophoenician woman and the centurion were secondary tasks in Jesus' overall public ministry to Jews, and Samaritans are part Jewish.

Now, Peter begins to understand. All those events in Jesus' ministry when He focused on the marginalized, the "unclean," and the foreigners weren't additions to or distractions from the mission—they are the mission. Jesus came for Jews and Samaritans and Gentiles.

When a passage says the speaker "opened his mouth" (Matthew 5:2), it means they said something important. The next sentence Peter says is the start of the international church.