It is impossible to know one way or another.
The Nygard Incident
I wouldn't call Russia Today a reliable source, nor would various Western media, both mainstream and political. But let's leave that aside for now and take the question and incident by themselves.
The incident was a real incident involving one Kahler Nygard, who was not on the No Fly List (after all, he flew), but on the Selectee List, a different list of people who must undergo additional screening before boarding a flight (SSSS screening). Many reports of celebrities being put on the No Fly List are actually cases of them being mistaken for people on the Selectee List, some notable cases including Sen. Ted Kennedy, musician Cat Stevens, and Fox News contributor Steve Hayes. The TSA agent attempted to subject Nygard to another screening after his flight, which makes no sense, and which I'm pretty certain is outside of a TSA screener's power.
Like the No-Fly List, the criteria for being on the Selectee List are not published. But a 2013 New York Times report notes that the TSA has been pre-screening domestic passengers by searching databases of car registrations, employment information, and so on. So while Nygard might have been flagged for online activities, he might also have been caught up because he is an activist, and his father is something of a community rabble-rouser; the local police may have submitted his name for monitoring as a troublemaker. To be clear, I do not think this would be any more justifiable, nor that the attempt to search him was in any way justifiable. But it is a more plausible, less conspiratorial, and frankly less conceited explanation.
Nomination and Selection Process
The No Fly List (which bans individuals from boarding commercial aircraft in or into the United States) and Selectee List (or SSSS list, which subjects individuals to additional screening at airport security checkpoints) are administered since 2003 as subsets of the Terrorist Watchlist (officially, the Terrorist Screening Database, TSDB), and managed since 2004 by a multi-agency office known as the Terrorist Screening Center. After the "underwear bomb" attempt in 2009, early in the Obama administration, the list was considerably expanded.
But the actual criteria or formulae the TSC uses to list people in the TSDB are not published. Whether you look at the March 2010 Testimony before the House Judiciary Committee by TSC Director Timothy Healy, or the May 2012 Terrorist Watchlist Study by the GAO, or the official response of the government in the Mohamed v. Holder case, there isn't even a hint.
In July 2014, The Intercept, an online publication backed by Glenn Greenwald, Laura Poitras, and Jeremy Scahill, published a critical report entitled “Blacklisted: he Secret Government Rulebook for Labeling You a Terrorist,” based in part on the March 2013 Watchlisting Guidance manual produced by the National Counterterrorism Center. They note that management of the list is weak, and rife with errors and sometimes-flimsy assumptions and profiling. But even they can't find anything that nails down specific "listable offenses"— perhaps because it's not just a matter of enumerating offenses.
In 2014, Christopher M. Piehota, Director of the TSC, testified before the House Subcommittee on Transportation Security outlining the process:
Nominations originate from credible information developed by our intelligence and law enforcement partners.… Federal departments and agencies submit nominations of known or suspected international terrorists to the National Counterterrorism Center (NCTC)…. Similarly, the FBI collects, stores, and forwards information to the TSC relating to domestic terrorists that may have connections to international terrorism.
… First, the facts and circumstances pertaining to the nomination must meet the reasonable suspicion standard of review. Second, the biographic information associated with a nomination must contain sufficient identifying data so that a person being screened can be matched to or disassociated from another watchlisted individual.
Reasonable suspicion requires articulable facts which, taken together with rational inferences, reasonably warrant the determination that an individual “is known or suspected to be or has been engaged in conduct constituting, in preparation for, in aid of or related to terrorism and terrorist activities.” The reasonable suspicion standard is based on the totality of the circumstances in order to account for the sometimes fragmentary nature of terrorist information.…
Upon receiving the nomination, TSC personnel review the supporting information to assess sufficiency, including accuracy and timeliness. In particular, TSC personnel must make two determinations. First, they evaluate whether the nomination meets the reasonable suspicion standard for inclusion in the TSDB. This includes determining whether the derogatory information provided with the nomination meets the additional requirements for placing an individual on the No Fly or Selectee Lists. If a nomination involves a request that an individual be placed on the No Fly or Selectee Lists, the nomination must meet additional substantive criteria, above and beyond the “reasonable suspicion” requirement for TSDB nominations. Second, they consider whether the biographic information associated with a nomination contains sufficient identifying data so that a person being screened can be matched to or distinguished from a watchlisted individual on the TSDB.
Upon conclusion of the TSC’s review, TSC will either accept or reject the TSDB nomination. If a nomination is accepted, the TSC will create a TSDB record which includes only the “terrorist identifiers” (e.g., name, date of birth).
So, it is not a matter of do X and you will be put on the watchlist. There is a risk assessment, probably algorithmic at least in part. The calculus may be flawed in various ways, but we don't know, for obvious reasons: if the TSC were to publish such guidelines, actual terrorists (as well as child pornographers and drug dealers and everyone else trying to stay ahead of the law) would stop doing X and switch immediately to Y. And even if visiting unsavory websites is included in the mix, we wouldn't know visiting which websites, or how often, or with how much interaction would be required to get the red flag.
Does the government surveil what websites you visit? Yes. Ish. While some anecdotal claims have turned out to have other explanations, there is some monitoring of social network activity both by the Department of Justice and by the Department of Homeland Security and others (e.g. the IRS), as privacy watchdogs have discovered. The FBI and other agencies can also request that an ISP provide information about traffic logged for a certain IP address. And perhaps most infamously, as revealed by Edward Snowden, the NSA operates a program known as PRISM which collects data on activity at Apple, Google, Yahoo, Facebook, and other companies.
In practical terms, the government does not have unlimited resources, so a single visit to some dodgy website triggering alarms seems implausible. With something like 500,000 names in the TSDB, it's likely some participated in Bad Things on the Internet, but we wouldn't know how much to weigh which websites, what activity (posting hate? making threats? too many LOLcats?), for how often, for how long, from what locations, etc.
So as of this writing, I think the only answer that can be supported is that we don't know, and we're unlikely to ever learn.
One of the ironies here is that one of the former criticisms of the government was the lack of a centralized No Fly List before the September 11 attacks; even though some of the perpetrators were being monitored by the FBI, this information was never transmitted to the FAA. The criticism was repeated some years later at the 9/11 Commission hearings.
If you discover you are on the No-Fly List or SSSS list, or have otherwise been delayed or had screening problems and believe it to be erroneous, the Department of Homeland Security has created the Traveler Redress Inquiry Program (TRIP) system. You will be assigned a Redress Control Number for your case which can be included in airline reservations.