Acts 10:34 New English Translation

³⁴Then Peter started speaking:^[a] “I now truly understand that God does not show favoritism in dealing with people,^[b]

Acts 10:34 tn Grk “Opening his mouth Peter said” (a Semitic idiom for beginning to speak in a somewhat formal manner). The participle ἀνοίξας (anoixas) has been translated as a finite verb due to requirements of contemporary English style.
Acts 10:34 tn Grk “God is not one who is a respecter of persons,” that is, “God is not one to show partiality” (cf. BDAG 887 s.v. προσωπολήμπτης). L&N 88.239 translates this verse “I realize that God does not show favoritism (in dealing with people).” The underlying Hebrew idiom includes the personal element (“respecter of persons”) so the phrase “in dealing with people” is included in the present translation. It fits very well with the following context and serves to emphasize the relational component of God’s lack of partiality. The latter is a major theme in the NT: Rom 2:11; Eph 2:11-22; Col 3:25; Jas 2:1; 1 Pet 1:17. This was the lesson of Peter’s vision.

New English Translation (NET)