The main European countries involved with the exploration of "America" were Holland, England, France, and Spain for North America; Spain for Central America; and Spain, France, and Portugal for South America. Exploration in this question can be distinguished from settlement. Explorers tended to range over, map, and claim territory in the names of their sovereigns. Then the sovereigns defended their claims in various ways, including with armies and navies.
The Dutch and English mostly colonized the East Coast of North America, established settlements and an infrastructure of roads and waterways, and then slowly spread inland. The French and Spanish explorers ranged out over larger expanses of territories, building forts such as Fort Saint Louis to protect their claims. The Catholic Church, under Spain's oversight, established a series of missions such as San Antonio, San Diego, San Francisco, and Los Angeles over much of the Americas.
In terms of area magnitude of land explored and surveyed, Spain arguably took the lead in the European exploration of North, Central, and South America. Because Spain took over the Louisiana territory west of the Mississippi from France in 1762, Spain arguably led the rest of Europe in the exploration of the land area of the United States as well.