PS4 Pro 4K console is shipping with PS4 Firmware version 3.70 which according to Darthsternie's Archive wasn't released as an update, with several reports and a confirmation HERE by @kallek.
Following Sony's Sharefactory Update 2.0 and despite the recent Updating PS4 Firmware Guide, today JarodMadePro made a video demonstrating that in order to use the Web browser the PlayStation 4 Pro Firmware needs to be updated first.
Hit it up below, and from the video's caption: PS4 PRO 3.70 - UPDATE TO ACCESS BROWSER
Just a video to show people on PSXHax that you need to update the PS4Pro to even access the browser.
@Derf321 also notes the following tip, to quote: "using DNS redirection, you could send the PS4 to any URL you want by loading the Users Manual page."
Cheers to @VultraAID in the PSXHAX Shoutbox for passing along the news earlier today!