Probably not, but this article in The Times suggests he was thinking along similar lines last year:
WASHINGTON — The deputy attorney general, Rod J. Rosenstein, suggested last year that he secretly record President Trump in the White House to expose the chaos consuming the administration, and he discussed recruiting cabinet members to invoke the 25th Amendment to remove Mr. Trump from office for being unfit.
Mr. Rosenstein made these suggestions in the spring of 2017 when Mr. Trump’s firing of James B. Comey as F.B.I. director plunged the White House into turmoil. Over the ensuing days, the president divulged classified intelligence to Russians in the Oval Office, and revelations emerged that Mr. Trump had asked Mr. Comey to pledge loyalty and end an investigation into a senior aide.
Mr. Rosenstein was just two weeks into his job. He had begun overseeing the Russia investigation and played a key role in the president’s dismissal of Mr. Comey by writing a memo critical of his handling of the Hillary Clinton email investigation. But Mr. Rosenstein was caught off guard when Mr. Trump cited the memo in the firing, and he began telling people that he feared he had been used.
Concern about Trump’s chaotic style — check.
Musings about the 25th Amendment — check.
I find the “feared he had been used” thing hard to believe. How could he expect anything else from Trump? Also, this:
The president’s reliance on his memo caught Mr. Rosenstein by surprise, and he became angry at Mr. Trump, according to people who spoke to Mr. Rosenstein at the time. He grew concerned that his reputation had suffered harm and wondered whether Mr. Trump had motives beyond Mr. Comey’s treatment of Mrs. Clinton for ousting him, the people said.
Rosenstein can’t possibly be that naive. Recall that his memo excoriated Comey for sandbagging Clinton, even though that aspect is often overlooked in accounts of Comey’s firing. Rosenstein cannot possibly believe that the “lock her up” guy gave a flying shit about Clinton being treated unfairly.
Rosenstein issued a statement calling the story “inaccurate” and “factually incorrect.” Anyhoo, interesting, and it’ll probably drive Spanky even battier.
ETA: Someone on Twitter:
this rosenstein story is a) fucking stupid b) clearly planted by someone in the admin, either to give a pretext to fire him or because of brett kavanaugh, idk. it's from may 2017! why now? c) boring. change the channel marge
— libby watson (@libbycwatson) September 21, 2018
Maybe she’s right.
ETA 2: The fix is definitely in:
Shocked!!! Absolutely Shocked!!! Ohhh, who are we kidding at this point? No one is shocked that these guys would do anything in their power to undermine @realdonaldtrump.
Rosenstein Suggested He Secretly Record Trump and Discussed 25th Amendment https://t.co/ZTEwS8E1iJ
— Donald Trump Jr. (@DonaldJTrumpJr) September 21, 2018
Dim Son dutifully chiming in after the story drops definitely confirms it’s a load of horse shit.