Development of archaeology as a field of discipline, is a very long process. You can see some important dates of archaeological events here: http://archaeology-faq.blogspot.com/2007/02/how-did-archaeology-start-as-field-of.html
It goes back to 16th century. But for me, modern archaeology is much more than earlier "antiquities collection making". So, modern archaeology started at the end of the 19th century (1880-90s). At these times, many sites excavated and the aim of archaeological investigation changed. Before, people collected artifacts just for pleasure (or treasure !) but at the end of 19th century, people started to see archaeology as an answers to their questions about the past.
About this long process, we cannot say a certain country as a homeland of archaeology. It is the continent of Europe. But people in many countries individually developed ideas about archaeology. France is the home of Paleolithic archaeology, Germany is well known about old languages etc. Every European country had its own interests about archaeology. We must say, every country has its own archaeology tradition. So we cannot say a country as the origin of archaeology. But we can say western Europe is the origin.