BRUSSELS (Reuters) - The Greek government should take a leaf out of South Korea's book and invest in better networks and getting more people online, a study which measures how the Web raises GDP levels said on Friday.

While many European countries make noticeable gains in their economic output as the Internet makes shopping cheaper and helps new businesses grow quickly, Greece's heavily indebted economy is far behind, research by Boston Consulting Group found.

South Korea is number one in the world for online retail, high Internet penetration and network coverage combined, an index compiled by the consultancy shows.

In 2010 Greece raised just 1.2 percent of gross domestic product (GDP) from the Internet, compared to 7.3 percent in South Korea.

Just under half of all households in Greece have fixed broadband access and 67 percent of Greeks use their mobiles to access the Internet. But only 12 percent of Greeks shop online, compared to a 40 percent European average, Boston Consulting said.

Its study, sponsored by Web search giant Google, said the Greek government should move quickly to get its businesses online by making company websites and online tax forms compulsory, getting consumers to buy online and giving incentives to network providers to expand their services.

Though 80 percent of Greek businesses have broadband access, a far lower number actually use the Web to sell their products to a wider audience. And for consumers, a majority will spend time discovering products online, but go to a shop to buy them.

Even Greek entrepreneurs who would like to tap foreign consumers online can find that red tape can prevent them from getting such businesses up and running.

The Greek Ekathimerini newspaper cited the case of Fotis Antonopoulos, who spent 10 months haggling with tax officials, the municipal service, doctors, health inspectors, the fire service and banks before he could sell olive oil online.

Antonopoulos eventually decided to let PayPal, the U.S.-based online payments company, handle the transactions, after local banks refused to handle payments if the website was not in Greek, the newspaper reported.

Boston Consulting cited Britain as an example of a country that had felt the benefits of boosting the role of the Web in its economy, saying high credit card usage there helped lift the Internet's total contribution to GDP to 8.3 percent in 2010.

(Editing by Catherine Evans)