The 38-year-old Argentine superstar striker was tightly marked throughout the match, a first leg showdown in the round of 16 in the North American club showdown.
Inter Miami will host the second leg next Wednesday.
Messi moved within one goal of the milestone 900th career tally after netting his 80th goal for Miami, and 899th of his career, last Saturday in an MLS triumph over DC United.
The eight-time Ballon D'Or winner, who turns 39 in June, scored 672 goals for Barcelona earlier in his career and 32 at Paris Saint-Germain, with a further 115 international goals for Argentina.
Rain arrived after a scoreless first half but defenders still dominated.
Messi was denied in the 57th minute by Nashville goalkeeper Brian Schwake's save on a right-footed shot from the right side of the box.
Inter Miami keeper Dayne St. Clair denied a header by Nashville's Reed Baker-Whiting in the 65th minute off a Cristian Espinoza cross to keep the match scoreless.