One might use one of several expressions suggesting a solid turning to a liquid. For instance, 'The/Their awkwardness melted away', in this example from Rudolf Vrba's book 'I Escaped From Auschwitz', Page 192 (https://books.google.com.au/books?isbn=1861059272):
Again the concept of a solid blockage melting away, and fluidity being restored to human relations is conveyed by the expression, 'The/Their awkwardness dissolved', in this example from Erika Robuck's book, 'Call Me Zelda' (https://books.google.com.au/books?isbn=1101614153):
The first example suggesting 'melting' not only suggested the solid impasse being replaced by a flow of conversation, but also that the agent of the change was a a 'warming' that occurred between the parties. It is interesting that we use so many metaphors of cold and hot to describe relationships and situations. One might imagine that the moment before the thaw could have been a 'frosty silence' or a 'chilly stare'
Taking this calorific view of social relations might also lead to the expression 'awkwardness thawed' in this example from Arthur de Carle Sowerby's 'The China Journal: Volume 30' from 1939:
All of this leads back to a concluding observation that 'Melting the ice' is a much gentler phrase suggesting a warming of a relationship than the more abrupt 'Breaking the ice'. The latter usage suggests an incident external to the parties, transient, incidental and possibly contrived in character rather than one driven by the growing warmth of the relations between the two parties as is conveyed by the first mentioned usage.
If we assume (in the care free way that Googles Ngrams seems to encourage) that references to Ice-breaking ships and global warming cancel each other out, then the relative popularity of 'Breaking the ice' and 'Melting the ice' is suggested in the following Ngram:
'Breaking the ice' is of course a expression with deep roots and rich usage, but that is far outside, and far larger a discussion, than this question and answer allows for today.