There is probably nothing in North America directly comparable to the pilgrimage trails in the Old World; the continent was too recently settled and too lacking in sites associated with Jesus and the Apostles (pace Latter-Day Saints) for such a culture to develop, and U.S. society was suspicious of, and sometimes outright violent against, Catholic and Orthodox worship until well into the 20th century.
There are innumerable pilgrimage destinations (though few of international fame), notably shrines devoted to the patron saint of some nationality or ethnic group. The customary way to visit them is simply to drive there, or to take a bus. There might be points of interest close together that can be walked; for instance, the National Shrine of Our Lady of Martyrs is about six miles (~10 km) down the Eric Canal Trail from the National Saint Kateri Tekakwitha Shrine in upstate New York; there might even be a suggested walk, as with the Painted Churches of Texas. But there is no tradition associated with them; most visitors will think of it as a self-guided tour, at best.
In recent years, there have been an increasing number of organized walks, hikes, processions, marches, etc. with such destinations in mind. By and large, however, these are best thought of as pilgrimage events, not pilgrim's ways. They follow public roads or trails, and have no albergues or waypoint shrines or any other special services for pilgrims. Indeed, you may have to go considerably off-trail for food, water, and shelter.
A classic example is the Walking Pilgrimage to Doylestown to visit the National Shrine of Our Lady of Czestochowa, sponsored by SS. Peter & Paul Church in Great Meadows, New Jersey annually since 1987. In theory, you could follow their itinerary on your own. But this is not an established, year-round route; it is a selection of public trails going through public parkland. You are entirely on your own for camping, eating, and other arrangements.
More common are single-day walking events. In Washington, D.C., for example, the Archdiocese has organized a Lenten 7-Church Walk annually since 2003, in loose imitation of the traditional walks to the seven Pilgrim Churches of Rome. At each church, the group receives some background on the parish and the worship space and prays a different form of prayer (e.g. Divine Mercy Chaplet, Rosary, Litany of Loredo, etc.). But the itinerary changes year to year, and again, there is no dedicated path, just public sidewalks and streets.
The nearest thing I could find to an effort to develop a full-fledged pilgrimage trail is the nascent Camino del Norte a Chimayó, which would follow a 115-mile (185 km) route over seven days from San Luis, Colorado to El Santuario de Chimayo in New Mexico, the most famous shrine in the Western U.S. Since 2012, a group has made the journey once a year, but as elsewhere, there is no infrastructure to serve pilgrims. It is at best a route, not a trail, and very much in its infancy.
[T]he Camino del Norte a Chimayó wanders along roadsides and forest trails through the sparsely-populated and desert-like sagebrush flats, rolling hills, and mountains of southern Colorado and northern New Mexico. One walks for as long as 20 miles (32 km) without a place to refill water bottles, purchase lunch or a snack, or even find a shady spot to rest for a few moments. Many of the villages and hamlets along the route have no commercial lodging or food markets, much less a café or restaurant. The only large town along the way is Taos, New
Mexico. Pilgrims along this route must therefore either camp out, carrying tents, food, and portable stoves, or if specifically planned in advance, stay overnight in church halls, sleeping on mats or air mattresses on hard floors.
Of course, there are many well-established secular trails, from famous treks like the John Muir Trail or the Appalachian Trail, or for the less alpinically-inclined, loops like the Freedom Trail. Many people have had spiritual experiences on the journey. The 2014 film Wild is a dramatization of a trail memoir of a woman who traveled solo for 1100 miles (1770 km) on the Pacific Crest Trail.