Nothng will accept .gvdesign files directly except Gravit Designer (and Klex).
For simple web animation without coding, have a look at some of the online options - I like the look of SVGator - uses SVG files (limited testing, but it works fine with some simple SVGs I tried from Gravit Designer). This is probably your best choice for simplicity.
For a more fully-featured offline tool, Google Web Designer has support for PNG and SVG import (not tested).
For character animation (people, animals, anthropomorphised objects), consider Synfig (accepts both SVG and PNG elements - PNGs work fine, haven’t tested it yet with SVG output from Gravit Designer).
Not as polished as something like Moho Pro nor as easy to use as Gravit Designer, but there is (in my opinion) no decent public Linux tool for 2D character animation (Blender might be getting there slowly, TupiTube still has a long way to go). OpenToonz might be an alternative to Synfig, still haven’t tried it yet.
For advanced motion graphics, have a look at Blackmagic Design Fusion - (accepts both SVG and PNG elements - haven’t tested it yet with any Gravit Designer output). Not as simple as Gravit Designer, but it has lots of high-end professional features.
Would love to see more good alternatives to Keyshape and Motion 5 - they might just convince me to get a Mac Mini for my next computer.
On Windows, offline, the best choice for non-coding web animation seems to be Saola Animate.
Also, Aphalina Animator (currently in beta) looks promising.