A lightly edited version of a previous answer:

The Dick Rowe quote about guitar groups being on their way out comes from Beatles manager Brian Epstein's autobiography, A Cellarful Of Noise, meant to explain why Decca didn't end up signing the Beatles:

Not to mince words, Mr Epstein, we don’t like your boys’ sound. Groups of guitarists are on their way out.

Rowe always denied it. Of course, by the time Rowe was in a position to need to deny it, the Beatles were the biggest band in the world (otherwise why would anybody care about the autobiography of their manager?) and so of course Rowe would deny it - nobody wanted to be the fool who denied the Beatles. On the other hand, it's fair to say that Epstein was motivated here to put words in Rowe’s mouth - the Beatles being so epochal, he might look like an insufficiently effective salesman as their manager if he couldn’t get them signed. And by the point in 1962 when the Beatles were in discussions with Decca, the Beatles were already a regional powerhouse with an international following thanks to their stays in Hamburg. They were a band who could command large fees thanks to the big crowds they got - and what’s more their manager was the manager of one of the largest and most influential record stores in the country (NEMS), and so someone they should want to keep happy. The London-based record companies really should have been interested in them at this point. Epstein had some explaining to do, if they weren’t...and the Beatles could definitely be mercenary about changing managers.

Firstly, it’s worth pointing out that the idea of the ‘guitar group’ in 1962 and the idea of the ‘guitar group’ after the Beatles are different things. In 1962, chances are that if you mentioned a ‘guitar group’ or 'group of guitarists', you were thinking of an instrumental guitar group. If he said the quote, Rowe was almost certainly thinking of The Shadows, who had once been Cliff Richard’s backing band before striking out on their own making instrumental music like Kon-Tiki’ and ‘Apache’. The Shadows’ last #1 single was early in 1963, so when the Beatles ended up not signing to Decca, if he actually said something of that ilk, he was not that far off with that prediction. Certainly the era of the instrumental guitar group - The Shadows, or The Tornadoes with ‘Telstar’, or American examples like Dick Dale and the Del Tones - would have looked like a brief fad to a music industry pro like Rowe. This is because it more or less was a brief fad, because such groups struggled commercially in the wake of the Beatles.

In contrast, the modern idea of the band a la the Beatles - the thing most readers are likely thinking of when they hear 'guitar group' - that is, a small-ish self-contained unit that did all the singing and all the instrument playing at the same time - was clearly a bit strange, as far as the British music industry in this era was concerned. Part of the struggle Brian Epstein had in getting the Beatles signed was that the Beatles' strength was as a unit that did all of the above, rather than a ‘singer and backing band’ combo or as an instrumental guitar group. As a result, many of the record executives Epstein talked to while trying to shop the band saw the Beatles as a risk because they were something of a new thing. Lewisohn argues that the Beatles were lucky they ended up paired with George Martin, who he argues was the only producer in London who would have let the Beatles essentially be themselves.

Secondly, it's also important to remember that The Beatles' story is usually presented as a fait accompli, because they ended up being so successful and influential - how could anyone have not seen that coming? How could anyone have not seen that the 1960s would just go nuts for 'guitar groups' or 'beat combos' or 'rock bands'? But Dick Rowe in 1962 didn't have a crystal ball, and he was ultimately trying to run a successful record label. And ultimately, it's clear that Dick Rowe took the idea of signing the Beatles seriously, recording an audition session in early 1962. But at this point, Lennon & McCartney were only budding songwriters - they hadn't yet written, say, 'She Loves You' or 'I Wanna Hold Your Hand'. It's clear they stepped up their game after discussions with George Martin and/or getting a sense of how the recording industry worked. But Rowe couldn't have known they were capable of that; most of the songs they presented at the audition were covers. Additionally, of course, the Beatles at this point had a drummer, Pete Best, who was adequate but not professional-level. Ultimately the Decca audition material on the Anthology 1 compilation doesn't give that much clue that the Beatles would be The Beatles.

With this in mind, other (non-George Martin) producers in London - if they had signed the Beatles - would have likely forced the Beatles into the ‘singer and backing band singing songs by professional songwriters’ mould that Brian Poole and the Tremeloes fell into. They would have picked either Lennon or McCartney as the lead singer (most likely Lennon, as McCartney had a track record of getting anxious at auditions and flubbing it), and having the rest be the backing band, likely replacing Best and maybe Harrison for the recording sessions. Perhaps they would have been successful in this kind of incarnation, but perhaps a group called John Lennon & The Beatles playing songs like 'How Do You Do It' would not have stopped the 'lead singer and guitar group' from going out of style.

In a February 3rd 1962 edition of Disc tracked down by Mark Lewisohn, Dick Rowe actually discussed his feelings about the rock and roll market in a trade newspaper:

I think there are about five separate markets for discs. We talk about teenagers but even that section is divided into three.

those between twelve and fourteen are not romantically inclined, and like the thumping rock style best.

those between fourteen and eighteen are romantically minded and enjoy the ballad style of people like Presley and Cliff Richard.

those between eighteen and twenty-two go for artists like Sinatra, and people older than that have other tastes.

Bands like The Beatles or Brian Poole and the Tremeloes (who Decca did sign after considering the Beatles) were likely what Rowe was thinking of with the ‘thumping rock style’ (though obviously to modern ears there's not that much thump compared to, say, 'Turn Up For What'); Decca signing the Tremeloes and saying in print that there was a market for a ‘thumping rock style’ is certainly evidence that Dick Rowe was aware that there was a market for rock & roll focused around guitars.

According to Mark Lewisohn’s book, the junior A&R executive Mike Smith wished to sign both The Beatles and Brian Poole and the Tremeloes, but Rowe said he could only sign one of the two bands. Smith agonised over the decision but instead went with the Tremeloes, because they were ultimately a more professional group, with good quality equipment, and who were based in London and thus easier to organise (and Smith was friends with Poole in the first place, given the small London pop music industry).

However, Lewisohn also presents evidence that Rowe tried to work out a compromise with Epstein, that while they wouldn’t produce the Beatles, they would allow a very young Decca A&R man, Tony Meehan (who was eighteen and maybe knew what the kids wanted), to produce the Beatles by private arrangement for a large fee, and Decca would put it out. There is some controversy, according to Lewisohn, as to whether the Decca tapes present on the Anthology (e.g., here) are these sessions made with Meehan, or are of the audition with Mike Smith; Lewisohn finds it more likely that there were no sessions with Meehan (not least because Meehan denies it) and the recordings are of the audition. It is perhaps likely that the Beatles/Epstein were offended by the Meehan offer, and refused it.

If anything, Dick Rowe is portrayed as being an Americophile compared to his UK A&R peers, and probably more receptive to rock and roll groups than most. There’s other music industry people of the era giving predictions of the next trends: Norrie Paramor, an EMI A&R man, had told Disc in 1961 that the next trend would be ‘waltz tunes’, according to Lewisohn, and a January 1962 edition of the NME included a list of six British ‘promising artists for 1962 stardom’: Pierce Rodgers, Lena Martell, Nick Villard, Anita Harris, Iain Gregory and Suzy Cope - none of whom ended up being half as big as the Beatles.

Did Rowe tell Epstein that 'groups of guitarists' were on the way out? I mean, maybe. I personally think it's more likely that he thought there was a place for groups like what he understood the Beatles to be, as he mentioned as such in print around the time - and he approved the signing of Brian Poole and The Tremeloes, who were initially very much in the mould of Buddy Holly and the Crickets (e.g., see their 1962 single 'Blue', released fairly soon after they were signed), and apparently he would have signed the Beatles if Mike Smith had chosen them over the Tremeloes. And, chances are, Rowe most likely thought that groups like the Shadows were on their way out (because they were). But Epstein had his own motivations for saying the quote, and Decca didn’t quite fully reject the Beatles in quite the way that Epstein portrayed it in print.

References: Mark Lewisohn’s The Beatles All These Years: Tune In (Extended Special Edition, Part Two)