Some inaccuracies here...
State universities in the US do not offer full rides to international students unless they receive NCAA (sport) scholarships.
Ivy League schools do not offer NCAA scholarships.
Some state universities offer academic merit scholarships. They do not offer financial aid (basically a waiver of varying proportions of the cost of attendance), and international students are not eligible for federally funded loans or grants.
U Iowa, for example, offers a limited number of merit scholarships to international students, valued between $2k to $15k pa. The maximum award at Iowa would only cover a fraction of the international cost of attendance, which ranges from approximately $50k to $54k depending on school.
The University of Idaho offers two scholarships worth half the cost of tuition (not cost of attendance) to international students annually.
There are a select few extremely selective private universities and liberal arts colleges that offer need blind admission and will cover the full demonstrated financial need of international students.