Since the verbs "to browse" and "to surf" both exist in English with reference to moving from one site to another and another and another in the Internet, the use of the neuGerman verb "surfen" for "to browse" would be very dependent on the context and possibly not right.
Originally "to browse" meant "schmökern", "umschauen", "umsehen", "durchblättern", "durchforschen", or "blättern", it meant an unhurried, relaxed perusal of a book, many books or magazines in a book shop or news agent's, or clothes in an apparel shop. In contrast, "to surf" meaning "surfen" meant a much faster, more hectic, agile, urgent activity.
The difference has usually been retained in the Internet meanings of the two verbs. An Internet "surfer" leaps from site to site with much agility and speed like a "surfer" on a board moves over the water. To explain the way an exuberant teenager surfs the Internet, one could never say "s/he is browsing the Internet". When I am moving methodically from site to site looking for word usage in the Internet, that same teenager would laugh if I said I was "surfing" the net.
Unfortunately for the translator, the Internet surfer and the person browsing through the Internet both use the same tool, a "browser" and the person might be a "surfer" but almost never a "browser".
For "to browse", you could certainly use "browsen", "umschauen", "durchforschen", "umsehen", "durchforsten", "durchblättern", or "blättern" depending on your context but "surfen" might imply too much speed and "schmökern" might imply too little speed.
As Alexander mentioned above, the other neuGerman verb "browsen" seems to have been quite clearly "eingeGermaned". Even when the ".de" domain, as well as the language "German", is added to the query, there are still nearly 10,000 hits for "browsen".
Local time: 11:03
Native speaker of: English
PRO pts in pair: 390