It is not only by context that the truth of Scripture is determined, but by comparing Scripture with Scripture, which is what I did. The devil can say, 'It is written,' but the Lord Jesus replies, 'It is written again' (Matthew 4:6-7).
But the context of John 6:37 is believing, which is equated with coming. "I am the bread of life. He who comes to Me shall never hunger, and He who believes in Me shall never thirst' (v.35). To come to Christ is to believe or trust in Him for salvation and so it is in v.37.