The Gregorian Calendar was created so that annual astronomical events, specifically the vernal equinox (used to determine when Easter is), would on average keep their places in the calendar year over time. It is the best official approximation to the definition of the tropical year, which is defined as "the length of time that the Sun takes to return to the same position in the cycle of seasons". Because this calendar describes 97 leap years out of every 400 years, it defines the average year as exactly 365.2425 solar days, or exactly 365 days, 5 hours, 49 minutes, and 12 seconds.
However, the mean tropical year is in reality about 365 days, 5 hours, 48 minutes, and 45 seconds, or 27 seconds shorter.
Because the Gregorian Calendar is based on the tropical year, the calendar dates of the year will keep up with the solstices and equinoxes, and thus the seasons. If this calendar were exactly the length of the tropical year, then the calendar would keep the vernal (northward) equinox around March 20th for all time.
But because of the slight inaccuracy, it will take about 3,200 years (60 s/min * 60 min/hr * 24 hr/day / 27 s/year) for these 27 seconds to add up every year to be 1 full day, and that will result in the solstices and equinoxes marching backwards in the calendar by 1 day every 3,200 years or so, depending on the accuracy of the 27 seconds difference. This very slow shift is due to the slight inaccuracy in the Gregorian calendar in, on average, matching the tropical year, not because of the precession of the equinoxes.
3,200 years from now, if the Gregorian Calendar is still used, the date of the vernal (northward) equinox will be on average one day earlier in March. The precession of the equinoxes will still occur, so the Earth's axis tilt will be significantly different from today. The Earth will be at a noticeably different position with respect to the Sun on the vernal (northward) equinox from where it is today, in 2014, on the vernal (northward) equinox, but it will still be in March.
This inaccuracy may very slowly increase over time, because according to the same Wikipedia page for the tropical year, the tropical year is very slowly getting shorter, and the mean solar day is even more slowly getting longer. But for 10,000 years to come, the Gregorian Calendar will keep the vernal (northward) equinox in March, even if it slowly shifts earlier in the month.
This is in contrast to the scenario that you imply, where the calendar date would correspond to the relative position of the Earth in its orbit around the Sun. That describes the sidereal year, the time taken for the Sun to reach the same spot in the sky relative to the stars, which is 365 days, 6 hours, 9 minutes, and 10 seconds. A sidereal calendar would explain why you might think that precession would cause the dates of equinoxes and solstices to change in the calendar year. That would result in a shift in the calendar of one full month in 1/12th the cycle length of the precession of the equinoxes, or about 1 full month about every 2,000 years.