Yes, the order of the Ten Commandments is significant. Jesus told us plainly that "the first" of the commandments, which in Greek has dual meaning--applicable both to sequence and to importance--was that connected with not having any other gods.
And Jesus answered him, The first of all the commandments is, Hear, O
Israel; The Lord our God is one Lord: And thou shalt love the Lord thy
God with all thy heart, and with all thy soul, and with all thy mind,
and with all thy strength: this is the first commandment. (Mark 12:29,
Essentially, Jesus identifies the Hebrew "Shema" (starting in Deuteronomy 6:4) as referencing "the first commandment." If one accepts that there is only one God, then one cannot have "any other gods" before Him.
If one truly keeps the first commandment, the others will be easy to follow.
Consider the last six of the ten, in their order.
- Honoring Father and Mother
- Not committing murder
- Not committing adultery
- Not stealing
- Not bearing false witness
- Not coveting
Apart from the first of these, most would agree that they are in descending order of importance. I'd rather that someone stole from me than that he or she murdered me. Even adultery seems to be of lesser magnitude than murder. The fact that honoring one's parents is listed before any of these others should stimulate some thought. Apparently, God sees that commandment as being of great importance.
In terms of its length, the fourth commandment holds the greatest attention of any of the Ten. It is also the only one that starts with "remember." God, who sees the future from the beginning, knew that it would be among the first which people would want to forget. Today, most ignore it, or try to claim it was nailed to the cross and no longer applies--despite having been written in unchangeable stone by the finger of God Himself.
Still, what benefit would there be to keeping the Sabbath if one has other gods? The first commandment is of greater importance.
The question is thought-provoking, and worth considering. The answer appears to be a clear "yes." God presented the commandments in the order of their importance, from the greatest to the least. To break one of the lesser commandments one must break those which come before it--they are all connected like a perfect chain.