Ecliptic line plot in a map
The ecliptic is a plane that is inclined 23.43 degrees (approximately, insert your more accurate value as needed). One position on the ecliptic is 0 hours Right Ascension, 0 degrees Declination (0,0). If you plot that on the Earth at the date and time you have, the rest of the ecliptic's path on the Earth can be calculated as follows:
$$\tan(23.43) \sin(longitude-longitude_{equatorcrossing}) = \tan(latitude)$$

where $longitude_{equatorcrossing}$ is the longitude where the ecliptic at 0 hours Right Ascension crosses the equator.

As a check, the latitude should be 23.43 degrees when the longitude is 90 degrees east of the (0,0) longitude point, and the latitude should be -23.43 degrees when the longitude is 90 degrees west of the (0,0) longitude point. In other words, if (0 Right Ascension, 0 Declination) corresponds to 15 degrees east longitude, then 105 east longitude is where the ecliptic is at the maximum northern latitude, and 75 degrees west longitude is where the ecliptic is at the maximum southern latitude.