No, you cannot play GameCube games on PS2. Zero compatibility. Here's the technical why:
| Component | GameCube (DOL-001) | PS2 (SCPH-30000+) |
|---|---|---|
| CPU | IBM PowerPC "Gekko" @ 486 MHz | MIPS "Emotion Engine" @ 294/299 MHz |
| GPU | ATI "Flipper" @ 162 MHz (TEV pipeline) | Sony "Graphics Synthesizer" @ 147 MHz (VRAM bandwidth monster) |
| RAM | 24 MB 1T-SRAM + 16 MB DRAM | 32 MB RDRAM + 4 MB eDRAM (GS) |
| Media | 8cm MiniDVD (1.4 GB) | 12cm DVD-ROM (4.7 GB) / CD-ROM |
| Architecture | Unified memory, fixed-function TEV | Segmented memory, vector units (VU0/VU1) |
Bottom Line: Different planets. Keep each console for its exclusives.