Explanation
Yes. Australia's east-to-west width exceeds the Moon's diameter. Australia is almost 4,000 kilometers wide, while the Moon is approximately 3,480 kilometers across.
Geoscience Australia, the national geoscience agency, gives Australia's east-to-west span as almost 4,000 kilometers. NASA lists the Moon's radius as about 1,740 kilometers. Doubling that radius gives an approximate diameter of 3,480 kilometers, smaller than Australia's stated width.
This is a comparison of one linear dimension, not a claim that Australia is larger than the Moon in every sense. A country's span and a spherical body's diameter describe different geometries; the comparison does not establish that Australia has a greater total surface area.
Key takeaway
Australia is wider east to west than the Moon is across its diameter.