Newton did indeed have issues with Handel's Messiah, so much so that in 1784–85 he gave a series of 50 sermons at his own church (St. Mary Woolnoth) in response to the oratorio being played at Westminster Abbey as part of a commemoration festival of Handel's 100th birthday. His stated goal was to preach on the Scriptural passages found in the lyrics of the Messiah, though he also found ample opportunity to further his critique of the oratorio and its performance.

The complete collection of fifty sermons is available online in a number of formats, such as in this one-volume collection of Newton's works. Thankfully, there are also helpful summaries of the sermons – one by Robert Manson Myers, "Fifty Sermons on Handel's Messiah" (Harvard Theological Review, October 1946) and David W. Music, "John Newton's Sermons on Handel's Messiah" (American Organist, August 2003).

Dr. Music summarizes Newton's objections as follows:

1. The music of Messiah, though majestic and greatly admired, deflects attention from the scriptural text upon which it is based;
2. The audience that gathered to hear Messiah came for the wrong reasons; and
3. The place of its performance was inappropriate.

Messiah distracts from Scripture

One of Newton's concerns was that the impressive music of the Messiah drowned out any chance of valuing the actual words:

It is to be feared, that [...] many persons, who have received pleasure from the music of the Messiah, have neither found, nor expected, nor desired to find, any comfort from the words. (Sermon 1)

He asks those who have heard the Messiah to consider if they were "only captivated by the music," and treated the "great truths of God" found in the lyrics as "if they had no meaning" (Sermon 2). To Newton, the power of the words was being ignored because of the effect of the music:

The music of the Messiah is but an ornament of the words, which have a very weighty sense. This sense no music can explain, and when rightly understood, will have such an effect as no music can produce. (Sermon 3)

Messiah's audience

We know from contemporary observers, besides Newton, that the Handel Commemoration at Westminster Abbey was a frenzy—Myers describes the "frank worldliness of a brilliant scene," and the inability of church officials to control the audience (218). This fueled Newton's cricism of the event, and particularly those who attended it. They treated "the sufferings of the Son of God" as an "amus*m*nt" (Sermon 22); they would hear of "his passion and his kingdom" as a part of "musical entertainment, but upon no other occasion" (Sermon 11).

Newton draws an analogy between the Messiah's audience and a group being tried for high treason. A guilty verdict is inevitable, but instead of seeking mercy from the king, they are "wholly taken up with contriving methods of amusing themselves" so as to maintain "as much chearfulness as possible." Their preferred form of entertainment is music, and they especially like a work that describes their trial, judge, and sentence, and even, upon its receipt, incorporates the wording of the offer of pardon sent by the king. "The fearful doom awaiting them [...] is sung for their diversion"; they focus on their entertainment and take no step toward accepting the king's mercy. (Sermon 4)

The seriousness of the subject of the Messiah makes its use as a "fashionable amus*m*nt" even worse:

If the far greater part of the people who frequent the Oratorio, are evidently unaffected by the Redeemer's love, and uninfluenced by his commands, I am afriad, it is no better, than a profanation of the name and truths of God, a crucifying the Son of God afresh. (Sermon 50)

The place of performance

Newton also objected to the Handel Commemoration taking place at Westminster Abbey, in part because audiences considered performances of the Messiah to be "equally acceptable, whether performed in a church or in the theatre" (Sermon 50). But especially egregious was that praise of Handel was allowed to supersede praise of God in worship services:

Mr. Handel [...] has been commemorated and praised, many years after his death, in a place professedly devoted to the praise and worship of God; yes (if I am not misinformed), the stated worship of God in that place was suspended for a considerable time, that it might be duly prepared for the commemoration of Mr. Handel. (Sermon 4)

Summary

Kevin Twit is thus more or less correct; Newton did indeed object to Messiah, and certainly would not have allowed it to be played in his church (though it's not clear that that was ever proposed). Still, Newton was not so much concerned with the "musical style" of Messiah as with the way it diverted attention from and made entertainment out of the Gospel.