No, the opposite is true.
Firstly, Dr. Aric Sigman himself has disclaimed the contents of the article:
Dr Sigman has published other papers. Well Connected?: The Biological
Implications of 'Social Networking', is published in The Biologist,
Vol 56(1), the journal of the Society of Biology.
Note: This paper has been misrepresented by some news reports,
websites and bloggers as claiming that social networking causes cancer
or disease. This is not true. The paper addresses the extent to which
time online may be displacing face-to-face contact, and that lack of
social connection is associated with physiological changes, increased
incidence of illness and higher premature mortality.
—Aric Sigman website
More in general, even his claims on his website have been heavily criticised by Dr. Ben Goldacre on his blog, where he provides three counter-example studies:
1. Caplan SE published a paper in 2007 entitled: “Relations among loneliness, social anxiety, and problematic Internet use.” Dr Sigman did not quote this paper in his article. Why not? “The results support the hypothesis that the relationship between loneliness and preference for online social interaction is spurious.”
2. Sum et al published a paper in 2008 with the title: “Internet use and loneliness in older adults“. Dr Sigman chose not to quote this paper. Why not? I don’t know, although it does contain the line “greater use of the Internet as a communication tool was associated with a lower level of social loneliness.”
3. Subrahmanyam et al published a paper in 2007 called “Adolescents on the net: Internet use and well-being.” It features the line “loneliness was not related to the total time spent online, nor to the time spent on e-mail”. Dr Sigman ignored it.
Another doctor, neurologist Vaughan Bell criticises Sigman on the Mindhacks blog:
For example, like one study that found that older adults who use the internet more report lower levels of loneliness, or this study in children that found internet use was associated with less loneliness, or this study that found no link in adolescents.
The article is quite clearly drivel if you spend more than 20 seconds on Google, but it seems to have been swallowed by most mainstream press outlets without question.
Finally, another paper completely disproving this claim that social media lead to loneliness (and consequently to disease), in fact finding that the opposite is true:
Based on the evidence presented in this article, it is plausible
to assume that online self-disclosure accounts for the positive
relationship between online communication and social connectedness.
—Social Consequences of the Internet for Adolescents by Patti M. Valkenburg and Jochen Peter, Amsterdam School of Communications Research ASCoR, University of Amsterdam, 2009.