The simple answer is no. Ice cream and nitrate are not homophonous with I scream and night rate.
If you go by tchrist’s comment above, it appears that some dialects of English pronounce the initial vowel in I scream [ɑɪ] and ice cream [ʌɪ] differently, but this is not universally applicable, and I would venture that this split is limited to a minority of dialects.
What distinguishes the two sets in all varieties of English is their syllable boundaries: morphological boundaries nearly always create phonemic syllable boundaries. In phonemic writing, indicating syllable boundaries with a period ‹.›, the two pairs look like this:
Ice cream /ˈaɪs.krim/ and nitrate /ˈnaɪ.treɪt/
I scream /aɪ.skrim/1 and night rate /ˈnaɪt.reit/
In English phonology, it is a well-known and universal fact2 that an unvoiced plosive /p t k/ is aspirated syllable-initially, but not in the syllable coda; in the case of /t/, this aspiration usually shows up as a slight affrication with little true aspiration. The initial aspiration applies only if the plosive is the first sound in the syllable, so initial /sp st sk/ are not aspirated. If an approximant (such as /r l j/) immediately follows an aspirated plosive, the aspiration of the plosive is carried over into the approximant, which is rendered devoiced.
If you notice, the first line in the quote above has syllable boundaries that cause the /k/ and the /t/ to be the first sound in the second syllable, while the second line has syllable boundaries that result in this not being the case. Therefore, in the first line, the plosives are aspirated and the following /r/ devoiced, while in the second, the plosive is unaspirated and the /r/ voiced; phonetically (in generic, Broadcast American):
Ice cream [ˈaɪs.kʰɹ̥ʷiːm] and nitrate [ˈnaɪ.tˢɹ̥ʷɛɪt]3
I scream [aɪ.skɹʷiːm] and night rate [ˈnaɪt.ɹʷɛɪt]
Since syllable-final /t/ is often reduced to an unreleased [t̚] or even just a glottal stop [ʔ], the second pair can be even further distinguished:
nitrate [ˈnaɪ.tˢɹ̥ʷɛɪʔ] vs. night rate [ˈnaɪt̚.ɹʷɛɪt̚] or [ˈnaɪʔ.ɹʷɛɪʔ]
1 I don’t mark the stress in this one since it’s a phrase, not a lexeme: it can be stressed on either syllable, depending on emphasis.
2 I’m not sure whether it applies in Indian English, and there are probably quite a few variants of African English where it doesn’t apply either; so understand ‘universal’ here to refer to ‘all dialects of British, Irish, Scottish, US, Canadian, South African, and Antipodean English’.
3 The sequence [tˢʰ] followed by the retroflex [ɹʷ] (or in American English more commonly [ɻʷ]) will normally merge somewhat, causing the /t/ to become retroflex as well. A more accurate phonetic notation would be [t͡ʂɹ̥ʷ], but for the sake of simplicity, I’ll ignore this here and just write [tˢɹ̥ʷ].